Abstract group $D_{28}.C_{18}$

• Label: 1008.20
• Order: \( 2^{4} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{5} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $504$
• Rank: $2$

Group information

Description:	$D_{28}.C_{18}$
Order:	\(1008\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 7 \)
Exponent:	\(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \)
Automorphism group:	$D_{28}:C_6^2$, of order \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) (generators)
Outer automorphisms:	$C_2\times C_6$, of order \(12\)\(\medspace = 2^{2} \cdot 3 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_7$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	7	8	9	12	14	18	21	24	28	36	42	72	84
Elements	1	29	2	6	58	6	28	42	12	6	210	12	56	36	252	12	168	72	1008
Conjugacy classes	1	2	2	2	4	1	2	6	4	1	12	2	4	3	12	2	12	6	78
Divisions	1	2	1	2	2	1	1	1	2	1	2	1	1	2	2	1	1	2	26
Autjugacy classes	1	2	2	2	4	1	1	2	4	1	4	2	2	2	4	2	2	4	42

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	36	27	0	12	0	3	0	78
Irr. rational chars.	4	5	2	6	1	5	3	26

Minimal Presentations

Permutation degree:	$24$
Transitive degree:	$504$
Rank:	$2$
Inequivalent generating pairs:	$144$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	144
Arbitrary	8	10	16

Constructions

Presentation:	$\langle a, b, c \mid b^{72}=c^{7}=[a,c]=1, a^{2}=b^{36}, b^{a}=b^{19}, c^{b}=c^{5} \rangle$
Permutation group:	Degree $24$ $\langle(2,3)(4,5)(6,7)(18,21)(19,23)(22,24), (17,18,20,22)(19,24,23,21), (2,4,6) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(Q_8:D_7)$ $\,\rtimes\,$ $C_9$	$(C_7\times Q_8)$ $\,\rtimes\,$ $C_{18}$	$Q_8$ $\,\rtimes\,$ $(C_7:C_{18})$	$(C_7:C_{72})$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{28}$ . $C_{18}$	$(C_3\times Q_8)$ . $F_7$	$C_3$ . $(Q_8:F_7)$	$(Q_8\times C_{21})$ . $C_6$	all 19

Elements of the group are displayed as words in the generators from the presentation given above.

Homology

Abelianization:	$C_{2} \times C_{18} \simeq C_{2}^{2} \times C_{9}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 288 subgroups in 60 conjugacy classes, 29 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $D_{14}:C_6$
Commutator:	$G' \simeq$ $C_{28}$	$G/G' \simeq$ $C_2\times C_{18}$
Frattini:	$\Phi \simeq$ $C_{12}$	$G/\Phi \simeq$ $C_2\times F_7$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{21}$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $D_{28}.C_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $C_3\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 1008: $D_{28}.C_{18}$

Order 504: $C_{28}:C_{18}$, $C_{28}.C_{18}$, $C_7:C_{72}$

Order 336: $C_{21}:\SD_{16}$

Order 252: $C_7:C_{36}$ x 2, $C_{14}:C_{18}$

Order 168: $Q_8\times C_{21}$, $C_7:C_{24}$, $C_3\times D_{28}$

Order 144: $C_9\times \SD_{16}$

Order 126: $C_7:C_{18}$, $C_7:C_{18}$

Order 112: $Q_8:D_7$

Order 84: $C_{84}$ x 2, $C_3\times D_{14}$

Order 72: $C_{72}$, $Q_8\times C_9$, $D_4\times C_9$

Order 63: $C_7:C_9$

Order 56: $D_{28}$, $C_7\times Q_8$, $C_7:C_8$

Order 48: $C_3\times \SD_{16}$

Order 42: $C_{42}$, $C_3\times D_7$

Order 36: $C_{36}$ x 2, $C_2\times C_{18}$

Order 28: $C_{28}$ x 2, $D_{14}$

Order 24: $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$

Order 21: $C_{21}$

Order 18: $C_{18}$ x 2

Order 16: $\SD_{16}$

Order 14: $C_{14}$, $D_7$

Order 12: $C_{12}$ x 2, $C_2\times C_6$

Order 9: $C_9$

Order 8: $Q_8$, $D_4$, $C_8$

Order 7: $C_7$

Order 6: $C_6$ x 2

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1008: $D_{28}.C_{18}$

Order 504: $C_{28}:C_{18}$, $C_{28}.C_{18}$, $C_7:C_{72}$

Order 336: $C_{21}:\SD_{16}$

Order 252: $C_7:C_{36}$ x 2, $C_{14}:C_{18}$

Order 168: $Q_8\times C_{21}$, $C_7:C_{24}$, $C_3\times D_{28}$

Order 144: $C_9\times \SD_{16}$

Order 126: $C_7:C_{18}$, $C_7:C_{18}$

Order 112: $Q_8:D_7$

Order 84: $C_{84}$ x 2, $C_3\times D_{14}$

Order 72: $C_{72}$, $Q_8\times C_9$, $D_4\times C_9$

Order 63: $C_7:C_9$

Order 56: $D_{28}$, $C_7\times Q_8$, $C_7:C_8$

Order 48: $C_3\times \SD_{16}$

Order 42: $C_{42}$, $C_3\times D_7$

Order 36: $C_{36}$ x 2, $C_2\times C_{18}$

Order 28: $C_{28}$ x 2, $D_{14}$

Order 24: $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$

Order 21: $C_{21}$

Order 18: $C_{18}$ x 2

Order 16: $\SD_{16}$

Order 14: $C_{14}$, $D_7$

Order 12: $C_{12}$ x 2, $C_2\times C_6$

Order 9: $C_9$

Order 8: $Q_8$, $D_4$, $C_8$

Order 7: $C_7$

Order 6: $C_6$ x 2

Order 4: $C_4$ x 2, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1008: $D_{28}.C_{18}$ ($C_1$)

Order 504: $C_{28}:C_{18}$ ($C_2$), $C_{28}.C_{18}$ ($C_2$), $C_7:C_{72}$ ($C_2$)

Order 336: $C_{21}:\SD_{16}$ ($C_3$)

Order 252: $C_7:C_{36}$ ($C_2^2$)

Order 168: $Q_8\times C_{21}$ ($C_6$), $C_7:C_{24}$ ($C_6$), $C_3\times D_{28}$ ($C_6$)

Order 126: $C_7:C_{18}$ ($D_4$)

Order 112: $Q_8:D_7$ ($C_9$)

Order 84: $C_{84}$ ($C_2\times C_6$)

Order 63: $C_7:C_9$ ($\SD_{16}$)

Order 56: $D_{28}$ ($C_{18}$), $C_7\times Q_8$ ($C_{18}$), $C_7:C_8$ ($C_{18}$)

Order 42: $C_{42}$ ($C_3\times D_4$)

Order 28: $C_{28}$ ($C_2\times C_{18}$)

Order 24: $C_3\times Q_8$ ($F_7$)

Order 21: $C_{21}$ ($C_3\times \SD_{16}$)

Order 14: $C_{14}$ ($D_4\times C_9$)

Order 12: $C_{12}$ ($C_2\times F_7$)

Order 8: $Q_8$ ($C_7:C_{18}$)

Order 7: $C_7$ ($C_9\times \SD_{16}$)

Order 6: $C_6$ ($D_{14}:C_6$)

Order 4: $C_4$ ($C_{14}:C_{18}$)

Order 3: $C_3$ ($Q_8:F_7$)

Order 2: $C_2$ ($D_{14}:C_{18}$)

Order 1: $C_1$ ($D_{28}.C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1008: $D_{28}.C_{18}$ ($C_1$)

Order 504: $C_{28}:C_{18}$ ($C_2$), $C_{28}.C_{18}$ ($C_2$), $C_7:C_{72}$ ($C_2$)

Order 336: $C_{21}:\SD_{16}$ ($C_3$)

Order 252: $C_7:C_{36}$ ($C_2^2$)

Order 168: $Q_8\times C_{21}$ ($C_6$), $C_7:C_{24}$ ($C_6$), $C_3\times D_{28}$ ($C_6$)

Order 126: $C_7:C_{18}$ ($D_4$)

Order 112: $Q_8:D_7$ ($C_9$)

Order 84: $C_{84}$ ($C_2\times C_6$)

Order 63: $C_7:C_9$ ($\SD_{16}$)

Order 56: $D_{28}$ ($C_{18}$), $C_7\times Q_8$ ($C_{18}$), $C_7:C_8$ ($C_{18}$)

Order 42: $C_{42}$ ($C_3\times D_4$)

Order 28: $C_{28}$ ($C_2\times C_{18}$)

Order 24: $C_3\times Q_8$ ($F_7$)

Order 21: $C_{21}$ ($C_3\times \SD_{16}$)

Order 14: $C_{14}$ ($D_4\times C_9$)

Order 12: $C_{12}$ ($C_2\times F_7$)

Order 8: $Q_8$ ($C_7:C_{18}$)

Order 7: $C_7$ ($C_9\times \SD_{16}$)

Order 6: $C_6$ ($D_{14}:C_6$)

Order 4: $C_4$ ($C_{14}:C_{18}$)

Order 3: $C_3$ ($Q_8:F_7$)

Order 2: $C_2$ ($D_{14}:C_{18}$)

Order 1: $C_1$ ($D_{28}.C_{18}$)

Series

Derived series

$D_{28}.C_{18}$

$\rhd$

$C_{28}$

$\rhd$

$C_1$

Chief series

$D_{28}.C_{18}$

$\rhd$

$C_{28}.C_{18}$

$\rhd$

$C_7:C_{36}$

$\rhd$

$C_7:C_{18}$

$\rhd$

$C_7:C_9$

$\rhd$

$C_{21}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$D_{28}.C_{18}$

$\rhd$

$C_{28}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_{12}$

$\lhd$

$C_3\times Q_8$

Character theory

Complex character table

See the $78 \times 78$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $26 \times 26$ rational character table.