Abstract group $C_{192}:C_2$

• Label: 384.8
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{6} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: \( 2^{5} \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $67$
• Trans deg.: $192$
• Rank: $2$

Group information

Description:	$C_{192}:C_2$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Automorphism group:	$C_{48}:C_2^3$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \) (generators)
Outer automorphisms:	$C_2\times C_{16}$, of order \(32\)\(\medspace = 2^{5} \)
Composition factors:	$C_2$ x 7, $C_3$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12	16	24	32	48	64	96	192
Elements	1	7	2	8	2	16	4	32	8	64	16	128	32	64	384
Conjugacy classes	1	2	1	3	1	6	2	12	4	24	8	32	16	32	144
Divisions	1	2	1	2	1	2	1	2	1	2	1	2	1	1	20
Autjugacy classes	1	2	1	2	1	2	1	2	1	2	1	2	1	1	20

Dimension	1	2	4	8	16	32	64
Irr. complex chars.	64	80	0	0	0	0	0	144
Irr. rational chars.	4	4	3	3	3	2	1	20

Minimal Presentations

Permutation degree:	$67$
Transitive degree:	$192$
Rank:	$2$
Inequivalent generating pairs:	$96$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	64
Arbitrary	2	4	34

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{192}=1, b^{a}=b^{161} \rangle$
Permutation group:	Degree $67$ $\langle(1,2,3,9,4,10,13,27,5,11,14,28,17,30,33,49,6,12,15,29,18,31,34,50,20,32,36,51,39,52,53,62,7,8,16,23,19,24,35,43,21,25,37,44,40,46,54,58,22,26,38,45,41,47,55,59,42,48,56,60,57,61,63,64) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 85 & 0 \\ 0 & 109 \end{array}\right), \left(\begin{array}{rr} 142 & 0 \\ 0 & 38 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{64}$ $\,\rtimes\,$ $S_3$	$C_{192}$ $\,\rtimes\,$ $C_2$	$C_3$ $\,\rtimes\,$ $\OD_{128}$	$(C_3:C_{64})$ $\,\rtimes\,$ $C_2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{32}$ . $D_6$	$D_6$ . $C_{32}$	$C_{96}$ . $C_2^2$	$(S_3\times C_8)$ . $C_8$	all 19

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

Homology

Abelianization:	$C_{2} \times C_{32} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 66 subgroups in 38 conjugacy classes, 25 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{32}$	$G/Z \simeq$ $D_6$
Commutator:	$G' \simeq$ $C_6$	$G/G' \simeq$ $C_2\times C_{32}$
Frattini:	$\Phi \simeq$ $C_{32}$	$G/\Phi \simeq$ $D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{192}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{192}:C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_2\times C_{32}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{128}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 384: $C_{192}:C_2$

Order 192: $S_3\times C_{32}$, $C_{192}$, $C_3:C_{64}$

Order 128: $\OD_{128}$

Order 96: $S_3\times C_{16}$, $C_{96}$, $C_3:C_{32}$

Order 64: $C_{64}$ x 2, $C_2\times C_{32}$

Order 48: $S_3\times C_8$, $C_{48}$, $C_3:C_{16}$

Order 32: $C_{32}$ x 2, $C_2\times C_{16}$

Order 24: $C_4\times S_3$, $C_{24}$, $C_3:C_8$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

Order 12: $D_6$, $C_{12}$, $C_3:C_4$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 6: $C_6$, $S_3$

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 384: $C_{192}:C_2$

Order 192: $S_3\times C_{32}$, $C_{192}$, $C_3:C_{64}$

Order 128: $\OD_{128}$

Order 96: $S_3\times C_{16}$, $C_{96}$, $C_3:C_{32}$

Order 64: $C_{64}$ x 2, $C_2\times C_{32}$

Order 48: $S_3\times C_8$, $C_{48}$, $C_3:C_{16}$

Order 32: $C_{32}$ x 2, $C_2\times C_{16}$

Order 24: $C_4\times S_3$, $C_{24}$, $C_3:C_8$

Order 16: $C_{16}$ x 2, $C_2\times C_8$

Order 12: $D_6$, $C_{12}$, $C_3:C_4$

Order 8: $C_8$ x 2, $C_2\times C_4$

Order 6: $C_6$, $S_3$

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 384: $C_{192}:C_2$ ($C_1$)

Order 192: $S_3\times C_{32}$ ($C_2$), $C_{192}$ ($C_2$), $C_3:C_{64}$ ($C_2$)

Order 96: $S_3\times C_{16}$ ($C_4$), $C_{96}$ ($C_2^2$), $C_3:C_{32}$ ($C_4$)

Order 64: $C_{64}$ ($S_3$)

Order 48: $S_3\times C_8$ ($C_8$), $C_{48}$ ($C_2\times C_4$), $C_3:C_{16}$ ($C_8$)

Order 32: $C_{32}$ ($D_6$)

Order 24: $C_4\times S_3$ ($C_{16}$), $C_{24}$ ($C_2\times C_8$), $C_3:C_8$ ($C_{16}$)

Order 16: $C_{16}$ ($C_4\times S_3$)

Order 12: $D_6$ ($C_{32}$), $C_{12}$ ($C_2\times C_{16}$), $C_3:C_4$ ($C_{32}$)

Order 8: $C_8$ ($S_3\times C_8$)

Order 6: $C_6$ ($C_2\times C_{32}$)

Order 4: $C_4$ ($S_3\times C_{16}$)

Order 3: $C_3$ ($\OD_{128}$)

Order 2: $C_2$ ($S_3\times C_{32}$)

Order 1: $C_1$ ($C_{192}:C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_{192}:C_2$ ($C_1$)

Order 192: $S_3\times C_{32}$ ($C_2$), $C_{192}$ ($C_2$), $C_3:C_{64}$ ($C_2$)

Order 96: $S_3\times C_{16}$ ($C_4$), $C_{96}$ ($C_2^2$), $C_3:C_{32}$ ($C_4$)

Order 64: $C_{64}$ ($S_3$)

Order 48: $S_3\times C_8$ ($C_8$), $C_{48}$ ($C_2\times C_4$), $C_3:C_{16}$ ($C_8$)

Order 32: $C_{32}$ ($D_6$)

Order 24: $C_4\times S_3$ ($C_{16}$), $C_{24}$ ($C_2\times C_8$), $C_3:C_8$ ($C_{16}$)

Order 16: $C_{16}$ ($C_4\times S_3$)

Order 12: $D_6$ ($C_{32}$), $C_{12}$ ($C_2\times C_{16}$), $C_3:C_4$ ($C_{32}$)

Order 8: $C_8$ ($S_3\times C_8$)

Order 6: $C_6$ ($C_2\times C_{32}$)

Order 4: $C_4$ ($S_3\times C_{16}$)

Order 3: $C_3$ ($\OD_{128}$)

Order 2: $C_2$ ($S_3\times C_{32}$)

Order 1: $C_1$ ($C_{192}:C_2$)

Series

Derived series

$C_{192}:C_2$

$\rhd$

$C_6$

$\rhd$

$C_1$

Chief series

$C_{192}:C_2$

$\rhd$

$S_3\times C_{32}$

$\rhd$

$C_{96}$

$\rhd$

$C_{48}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{192}:C_2$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_{32}$

$\lhd$

$C_{64}$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $144 \times 144$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $20 \times 20$ rational character table.