Abstract group $C_{63}:D_4$

• Label: 504.17
• Order: \( 2^{3} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{4} \cdot 3^{4} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \)
• Perm deg.: $20$
• Trans deg.: $252$
• Rank: $2$

Group information

Description:	$C_{63}:D_4$
Order:	\(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \)
Exponent:	\(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \)
Automorphism group:	Group of order \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) (generators)
Outer automorphisms:	$C_6^2$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_7$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	7	9	14	18	21	42	63	126
Elements	1	33	2	126	30	6	6	114	90	12	12	36	36	504
Conjugacy classes	1	3	1	1	3	3	3	9	9	3	3	9	9	57
Divisions	1	3	1	1	2	1	1	2	2	1	1	1	1	18
Autjugacy classes	1	3	1	1	2	1	1	2	2	1	1	1	1	18

Dimension	1	2	4	6	12	36
Irr. complex chars.	4	29	24	0	0	0	57
Irr. rational chars.	4	3	1	4	4	2	18

Minimal Presentations

Permutation degree:	$20$
Transitive degree:	$252$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	72
Arbitrary	4	6	14

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{14}=c^{18}=[a,c]=1, b^{a}=b^{13}c^{9}, c^{b}=c^{17} \rangle$
Permutation group:	Degree $20$ $\langle(2,3)(4,5)(6,7)(8,9)(10,11), (9,11)(13,15)(14,18)(16,17)(19,20), (8,10) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_{18}$ $\,\rtimes\,$ $D_7$	$D_{14}$ $\,\rtimes\,$ $D_9$	$C_{63}$ $\,\rtimes\,$ $D_4$	$C_9$ $\,\rtimes\,$ $(C_7:D_4)$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{42}$ . $D_6$	$C_{18}$ . $D_{14}$	$C_{14}$ . $D_{18}$	$C_{126}$ . $C_2^2$	all 9

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 370 subgroups in 48 conjugacy classes, 19 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_7\times D_9$
Commutator:	$G' \simeq$ $C_{126}$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $S_3\times D_7$
Fitting:	$\operatorname{Fit} \simeq$ $C_{126}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{63}:D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{42}$	$G/\operatorname{soc} \simeq$ $D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
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

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Classes of subgroups up to conjugation

Order 504: $C_{63}:D_4$

Order 252: $C_{63}:C_4$, $C_7\times D_{18}$, $C_9\times D_{14}$

Order 168: $C_{21}:D_4$

Order 126: $C_{126}$, $C_9\times D_7$, $C_7\times D_9$

Order 84: $C_{21}:C_4$, $S_3\times C_{14}$, $C_3\times D_{14}$

Order 72: $C_9:D_4$

Order 63: $C_{63}$

Order 56: $C_7:D_4$

Order 42: $C_{42}$, $C_3\times D_7$, $S_3\times C_7$

Order 36: $C_2\times C_{18}$, $D_{18}$, $C_9:C_4$

Order 28: $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

Order 24: $C_3:D_4$

Order 21: $C_{21}$

Order 18: $C_{18}$ x 2, $D_9$

Order 14: $C_{14}$ x 2, $D_7$

Order 12: $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 9: $C_9$

Order 8: $D_4$

Order 7: $C_7$

Order 6: $C_6$ x 2, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 504: $C_{63}:D_4$

Order 252: $C_{63}:C_4$, $C_7\times D_{18}$, $C_9\times D_{14}$

Order 168: $C_{21}:D_4$

Order 126: $C_{126}$, $C_9\times D_7$, $C_7\times D_9$

Order 84: $C_{21}:C_4$, $S_3\times C_{14}$, $C_3\times D_{14}$

Order 72: $C_9:D_4$

Order 63: $C_{63}$

Order 56: $C_7:D_4$

Order 42: $C_{42}$, $C_3\times D_7$, $S_3\times C_7$

Order 36: $C_2\times C_{18}$, $D_{18}$, $C_9:C_4$

Order 28: $C_2\times C_{14}$, $D_{14}$, $C_7:C_4$

Order 24: $C_3:D_4$

Order 21: $C_{21}$

Order 18: $C_{18}$ x 2, $D_9$

Order 14: $C_{14}$ x 2, $D_7$

Order 12: $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 9: $C_9$

Order 8: $D_4$

Order 7: $C_7$

Order 6: $C_6$ x 2, $S_3$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 504: $C_{63}:D_4$ ($C_1$)

Order 252: $C_{63}:C_4$ ($C_2$), $C_7\times D_{18}$ ($C_2$), $C_9\times D_{14}$ ($C_2$)

Order 126: $C_{126}$ ($C_2^2$)

Order 84: $C_3\times D_{14}$ ($S_3$)

Order 63: $C_{63}$ ($D_4$)

Order 42: $C_{42}$ ($D_6$)

Order 36: $D_{18}$ ($D_7$)

Order 28: $D_{14}$ ($D_9$)

Order 21: $C_{21}$ ($C_3:D_4$)

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

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

Order 9: $C_9$ ($C_7:D_4$)

Order 7: $C_7$ ($C_9:D_4$)

Order 6: $C_6$ ($S_3\times D_7$)

Order 3: $C_3$ ($C_{21}:D_4$)

Order 2: $C_2$ ($D_7\times D_9$)

Order 1: $C_1$ ($C_{63}:D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 504: $C_{63}:D_4$ ($C_1$)

Order 252: $C_{63}:C_4$ ($C_2$), $C_7\times D_{18}$ ($C_2$), $C_9\times D_{14}$ ($C_2$)

Order 126: $C_{126}$ ($C_2^2$)

Order 84: $C_3\times D_{14}$ ($S_3$)

Order 63: $C_{63}$ ($D_4$)

Order 42: $C_{42}$ ($D_6$)

Order 36: $D_{18}$ ($D_7$)

Order 28: $D_{14}$ ($D_9$)

Order 21: $C_{21}$ ($C_3:D_4$)

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

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

Order 9: $C_9$ ($C_7:D_4$)

Order 7: $C_7$ ($C_9:D_4$)

Order 6: $C_6$ ($S_3\times D_7$)

Order 3: $C_3$ ($C_{21}:D_4$)

Order 2: $C_2$ ($D_7\times D_9$)

Order 1: $C_1$ ($C_{63}:D_4$)

Series

Derived series

$C_{63}:D_4$

$\rhd$

$C_{126}$

$\rhd$

$C_1$

Chief series

$C_{63}:D_4$

$\rhd$

$C_9\times D_{14}$

$\rhd$

$C_{126}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{63}:D_4$

$\rhd$

$C_{126}$

$\rhd$

$C_{63}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 17 larger groups in the database.

This group is a maximal quotient of 13 larger groups in the database.

Character theory

Complex character table

See the $57 \times 57$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $18 \times 18$ rational character table.