Abstract group 1536.1699: $C_8\times C_{192}$

• Label: 1536.1699
• Order: \( 2^{9} \cdot 3 \)
• Exponent: \( 2^{6} \cdot 3 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{14} \)
• Perm deg.: $75$
• Trans deg.: $1536$
• Rank: $2$

Group information

Description:	$C_8\times C_{192}$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(192\)\(\medspace = 2^{6} \cdot 3 \)
Automorphism group:	$C_2.C_4^3.C_2^6.C_2$, of order \(16384\)\(\medspace = 2^{14} \)
Composition factors:	$C_2$ x 9, $C_3$
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 2$ (hence hyperelementary), and metacyclic.

Group statistics

Order	1	2	3	4	6	8	12	16	24	32	48	64	96	192
Elements	1	3	2	12	6	48	24	64	96	128	128	256	256	512	1536
Conjugacy classes	1	3	2	12	6	48	24	64	96	128	128	256	256	512	1536
Divisions	1	3	1	6	3	12	6	8	12	8	8	8	8	8	92
Autjugacy classes	1	2	1	3	2	4	3	3	4	2	3	1	2	1	32

Dimension	1	2	4	8	16	32	64
Irr. complex chars.	1536	0	0	0	0	0	0	1536
Irr. rational chars.	4	10	18	20	16	16	8	92

Minimal presentations

Permutation degree:	$75$
Transitive degree:	$1536$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	2	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{8}=b^{192}=1 \rangle$
Permutation group:	Degree $75$ $\langle(9,72,40,56,24,64,32,48,16,68,36,52,20,60,28,44,12,70,38,54,22,62,30,46,14,66,34,50,18,58,26,42,10,71,39,55,23,63,31,47,15,67,35,51,19,59,27,43,11,69,37,53,21,61,29,45,13,65,33,49,17,57,25,41) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 186 \end{array}\right), \left(\begin{array}{rr} 125 & 0 \\ 0 & 105 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{193})$
Direct product:	$C_8$ $\, \times\, $ $C_{64}$ $\, \times\, $ $C_3$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{96}$ . $C_{16}$	$C_{48}$ . $C_{32}$	$C_{32}$ . $C_{48}$	$C_{16}$ . $C_{96}$	all 72

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

Homology

Primary decomposition:	$C_{8} \times C_{64} \times C_{3}$
Schur multiplier:	$C_{8}$
Commutator length:	$0$

Subgroups

There are 164 subgroups, all normal (44 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_8\times C_{192}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_8\times C_{192}$
Frattini:	$\Phi \simeq$ $C_4\times C_{32}$	$G/\Phi \simeq$ $C_2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_8\times C_{192}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_8\times C_{192}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_4\times C_{32}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8\times C_{64}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1536: $C_8\times C_{192}$

Order 768: $C_4\times C_{192}$ x 2, $C_8\times C_{96}$

Order 512: $C_8\times C_{64}$

Order 384: $C_2\times C_{192}$ x 4, $C_4\times C_{96}$ x 2, $C_8\times C_{48}$

Order 256: $C_4\times C_{64}$ x 2, $C_8\times C_{32}$

Order 192: $C_{192}$ x 8, $C_2\times C_{96}$ x 4, $C_4\times C_{48}$ x 2, $C_8\times C_{24}$

Order 128: $C_2\times C_{64}$ x 4, $C_4\times C_{32}$ x 2, $C_8\times C_{16}$

Order 96: $C_{96}$ x 8, $C_2\times C_{48}$ x 4, $C_4\times C_{24}$ x 3

Order 64: $C_{64}$ x 8, $C_2\times C_{32}$ x 4, $C_4\times C_{16}$ x 2, $C_8^2$

Order 48: $C_{48}$ x 8, $C_2\times C_{24}$ x 6, $C_4\times C_{12}$

Order 32: $C_{32}$ x 8, $C_2\times C_{16}$ x 4, $C_4\times C_8$ x 3

Order 24: $C_{24}$ x 12, $C_2\times C_{12}$ x 3

Order 16: $C_{16}$ x 8, $C_2\times C_8$ x 6, $C_4^2$

Order 12: $C_{12}$ x 6, $C_2\times C_6$

Order 8: $C_8$ x 12, $C_2\times C_4$ x 3

Order 6: $C_6$ x 3

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1536: $C_8\times C_{192}$

Order 768: $C_4\times C_{192}$, $C_8\times C_{96}$

Order 512: $C_8\times C_{64}$

Order 384: $C_4\times C_{96}$ x 2, $C_2\times C_{192}$, $C_8\times C_{48}$

Order 256: $C_4\times C_{64}$, $C_8\times C_{32}$

Order 192: $C_2\times C_{96}$ x 2, $C_4\times C_{48}$ x 2, $C_{192}$, $C_8\times C_{24}$

Order 128: $C_4\times C_{32}$ x 2, $C_8\times C_{16}$, $C_2\times C_{64}$

Order 96: $C_2\times C_{48}$ x 3, $C_4\times C_{24}$ x 2, $C_{96}$ x 2

Order 64: $C_2\times C_{32}$ x 2, $C_4\times C_{16}$ x 2, $C_8^2$, $C_{64}$

Order 48: $C_2\times C_{24}$ x 3, $C_{48}$ x 3, $C_4\times C_{12}$

Order 32: $C_2\times C_{16}$ x 3, $C_4\times C_8$ x 2, $C_{32}$ x 2

Order 24: $C_{24}$ x 4, $C_2\times C_{12}$ x 2

Order 16: $C_2\times C_8$ x 3, $C_{16}$ x 3, $C_4^2$

Order 12: $C_{12}$ x 3, $C_2\times C_6$

Order 8: $C_8$ x 4, $C_2\times C_4$ x 2

Order 6: $C_6$ x 2

Order 4: $C_4$ x 3, $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1536: $C_8\times C_{192}$ ($C_1$)

Order 768: $C_4\times C_{192}$ ($C_2$) x 2, $C_8\times C_{96}$ ($C_2$)

Order 512: $C_8\times C_{64}$ ($C_3$)

Order 384: $C_2\times C_{192}$ ($C_4$) x 4, $C_4\times C_{96}$ ($C_2^2$), $C_4\times C_{96}$ ($C_4$), $C_8\times C_{48}$ ($C_4$)

Order 256: $C_4\times C_{64}$ ($C_6$) x 2, $C_8\times C_{32}$ ($C_6$)

Order 192: $C_{192}$ ($C_8$) x 8, $C_2\times C_{96}$ ($C_2\times C_4$) x 2, $C_2\times C_{96}$ ($C_8$) x 2, $C_4\times C_{48}$ ($C_2\times C_4$), $C_4\times C_{48}$ ($C_8$), $C_8\times C_{24}$ ($C_8$)

Order 128: $C_2\times C_{64}$ ($C_{12}$) x 4, $C_8\times C_{16}$ ($C_{12}$), $C_4\times C_{32}$ ($C_2\times C_6$), $C_4\times C_{32}$ ($C_{12}$)

Order 96: $C_{96}$ ($C_2\times C_8$) x 4, $C_{96}$ ($C_{16}$) x 4, $C_2\times C_{48}$ ($C_{16}$) x 2, $C_4\times C_{24}$ ($C_{16}$) x 2, $C_2\times C_{48}$ ($C_2\times C_8$), $C_2\times C_{48}$ ($C_4^2$), $C_4\times C_{24}$ ($C_2\times C_8$)

Order 64: $C_{64}$ ($C_{24}$) x 8, $C_2\times C_{32}$ ($C_2\times C_{12}$) x 2, $C_2\times C_{32}$ ($C_{24}$) x 2, $C_4\times C_{16}$ ($C_2\times C_{12}$), $C_4\times C_{16}$ ($C_{24}$), $C_8^2$ ($C_{24}$)

Order 48: $C_2\times C_{24}$ ($C_{32}$) x 4, $C_{48}$ ($C_{32}$) x 4, $C_{48}$ ($C_4\times C_8$) x 2, $C_{48}$ ($C_2\times C_{16}$) x 2, $C_2\times C_{24}$ ($C_4\times C_8$), $C_2\times C_{24}$ ($C_2\times C_{16}$), $C_4\times C_{12}$ ($C_2\times C_{16}$)

Order 32: $C_{32}$ ($C_2\times C_{24}$) x 4, $C_{32}$ ($C_{48}$) x 4, $C_4\times C_8$ ($C_{48}$) x 2, $C_2\times C_{16}$ ($C_{48}$) x 2, $C_4\times C_8$ ($C_2\times C_{24}$), $C_2\times C_{16}$ ($C_2\times C_{24}$), $C_2\times C_{16}$ ($C_4\times C_{12}$)

Order 24: $C_{24}$ ($C_{64}$) x 8, $C_2\times C_{12}$ ($C_2\times C_{32}$) x 2, $C_{24}$ ($C_2\times C_{32}$) x 2, $C_2\times C_{12}$ ($C_4\times C_{16}$), $C_{24}$ ($C_4\times C_{16}$), $C_{24}$ ($C_8^2$)

Order 16: $C_2\times C_8$ ($C_{96}$) x 4, $C_{16}$ ($C_{96}$) x 4, $C_{16}$ ($C_2\times C_{48}$) x 2, $C_{16}$ ($C_4\times C_{24}$) x 2, $C_2\times C_8$ ($C_2\times C_{48}$), $C_2\times C_8$ ($C_4\times C_{24}$), $C_4^2$ ($C_2\times C_{48}$)

Order 12: $C_{12}$ ($C_2\times C_{64}$) x 4, $C_2\times C_6$ ($C_4\times C_{32}$), $C_{12}$ ($C_8\times C_{16}$), $C_{12}$ ($C_4\times C_{32}$)

Order 8: $C_8$ ($C_{192}$) x 8, $C_2\times C_4$ ($C_2\times C_{96}$) x 2, $C_8$ ($C_2\times C_{96}$) x 2, $C_2\times C_4$ ($C_4\times C_{48}$), $C_8$ ($C_4\times C_{48}$), $C_8$ ($C_8\times C_{24}$)

Order 6: $C_6$ ($C_4\times C_{64}$) x 2, $C_6$ ($C_8\times C_{32}$)

Order 4: $C_4$ ($C_2\times C_{192}$) x 4, $C_2^2$ ($C_4\times C_{96}$), $C_4$ ($C_4\times C_{96}$), $C_4$ ($C_8\times C_{48}$)

Order 3: $C_3$ ($C_8\times C_{64}$)

Order 2: $C_2$ ($C_4\times C_{192}$) x 2, $C_2$ ($C_8\times C_{96}$)

Order 1: $C_1$ ($C_8\times C_{192}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1536: $C_8\times C_{192}$ ($C_1$)

Order 768: $C_4\times C_{192}$ ($C_2$), $C_8\times C_{96}$ ($C_2$)

Order 512: $C_8\times C_{64}$ ($C_3$)

Order 384: $C_2\times C_{192}$ ($C_4$), $C_4\times C_{96}$ ($C_2^2$), $C_4\times C_{96}$ ($C_4$), $C_8\times C_{48}$ ($C_4$)

Order 256: $C_4\times C_{64}$ ($C_6$), $C_8\times C_{32}$ ($C_6$)

Order 192: $C_{192}$ ($C_8$), $C_2\times C_{96}$ ($C_2\times C_4$), $C_2\times C_{96}$ ($C_8$), $C_4\times C_{48}$ ($C_2\times C_4$), $C_4\times C_{48}$ ($C_8$), $C_8\times C_{24}$ ($C_8$)

Order 128: $C_8\times C_{16}$ ($C_{12}$), $C_2\times C_{64}$ ($C_{12}$), $C_4\times C_{32}$ ($C_2\times C_6$), $C_4\times C_{32}$ ($C_{12}$)

Order 96: $C_2\times C_{48}$ ($C_2\times C_8$), $C_2\times C_{48}$ ($C_4^2$), $C_2\times C_{48}$ ($C_{16}$), $C_4\times C_{24}$ ($C_2\times C_8$), $C_4\times C_{24}$ ($C_{16}$), $C_{96}$ ($C_2\times C_8$), $C_{96}$ ($C_{16}$)

Order 64: $C_2\times C_{32}$ ($C_2\times C_{12}$), $C_2\times C_{32}$ ($C_{24}$), $C_4\times C_{16}$ ($C_2\times C_{12}$), $C_4\times C_{16}$ ($C_{24}$), $C_8^2$ ($C_{24}$), $C_{64}$ ($C_{24}$)

Order 48: $C_2\times C_{24}$ ($C_4\times C_8$), $C_2\times C_{24}$ ($C_2\times C_{16}$), $C_2\times C_{24}$ ($C_{32}$), $C_4\times C_{12}$ ($C_2\times C_{16}$), $C_{48}$ ($C_4\times C_8$), $C_{48}$ ($C_2\times C_{16}$), $C_{48}$ ($C_{32}$)

Order 32: $C_4\times C_8$ ($C_2\times C_{24}$), $C_4\times C_8$ ($C_{48}$), $C_2\times C_{16}$ ($C_2\times C_{24}$), $C_2\times C_{16}$ ($C_4\times C_{12}$), $C_2\times C_{16}$ ($C_{48}$), $C_{32}$ ($C_2\times C_{24}$), $C_{32}$ ($C_{48}$)

Order 24: $C_2\times C_{12}$ ($C_2\times C_{32}$), $C_2\times C_{12}$ ($C_4\times C_{16}$), $C_{24}$ ($C_2\times C_{32}$), $C_{24}$ ($C_4\times C_{16}$), $C_{24}$ ($C_8^2$), $C_{24}$ ($C_{64}$)

Order 16: $C_2\times C_8$ ($C_2\times C_{48}$), $C_2\times C_8$ ($C_4\times C_{24}$), $C_2\times C_8$ ($C_{96}$), $C_4^2$ ($C_2\times C_{48}$), $C_{16}$ ($C_2\times C_{48}$), $C_{16}$ ($C_4\times C_{24}$), $C_{16}$ ($C_{96}$)

Order 12: $C_2\times C_6$ ($C_4\times C_{32}$), $C_{12}$ ($C_8\times C_{16}$), $C_{12}$ ($C_2\times C_{64}$), $C_{12}$ ($C_4\times C_{32}$)

Order 8: $C_2\times C_4$ ($C_2\times C_{96}$), $C_2\times C_4$ ($C_4\times C_{48}$), $C_8$ ($C_{192}$), $C_8$ ($C_2\times C_{96}$), $C_8$ ($C_4\times C_{48}$), $C_8$ ($C_8\times C_{24}$)

Order 6: $C_6$ ($C_4\times C_{64}$), $C_6$ ($C_8\times C_{32}$)

Order 4: $C_2^2$ ($C_4\times C_{96}$), $C_4$ ($C_2\times C_{192}$), $C_4$ ($C_4\times C_{96}$), $C_4$ ($C_8\times C_{48}$)

Order 3: $C_3$ ($C_8\times C_{64}$)

Order 2: $C_2$ ($C_4\times C_{192}$), $C_2$ ($C_8\times C_{96}$)

Order 1: $C_1$ ($C_8\times C_{192}$)

Series

Derived series

$C_8\times C_{192}$

$\rhd$

$C_1$

Chief series

$C_8\times C_{192}$

$\rhd$

$C_8\times C_{96}$

$\rhd$

$C_4\times C_{96}$

$\rhd$

$C_4\times C_{48}$

$\rhd$

$C_2\times C_{48}$

$\rhd$

$C_2\times C_{24}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_8\times C_{192}$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_8\times C_{192}$

Supergroups

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

This group is a maximal quotient of 8 larger groups in the database.

Character theory

Complex character table

The $1536 \times 1536$ character table is not available for this group.

Rational character table

See the $92 \times 92$ rational character table.