Abstract group 1584.65: $C_{792}:C_2$

• Label: 1584.65
• Order: \( 2^{4} \cdot 3^{2} \cdot 11 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{3} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \cdot 5 \)
• Perm deg.: $28$
• Trans deg.: $792$
• Rank: $2$

Group information

Description:	$C_{792}:C_2$
Order:	\(1584\)\(\medspace = 2^{4} \cdot 3^{2} \cdot 11 \)
Exponent:	\(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \)
Automorphism group:	$C_{99}.C_{30}.C_2^4$, of order \(47520\)\(\medspace = 2^{5} \cdot 3^{3} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 4, $C_3$ x 2, $C_{11}$
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	9	11	12	18	22	24	33	36	44	66	72	88	99	132	198	264	396	792
Elements	1	199	2	200	2	400	6	10	4	6	10	8	20	12	20	20	24	40	60	40	60	80	120	240	1584
Conjugacy classes	1	2	1	3	1	4	3	5	2	3	5	4	10	6	10	10	12	20	30	20	30	40	60	120	402
Divisions	1	2	1	2	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	27
Autjugacy classes	1	2	1	2	1	2	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	27

Dimension	1	2	4	6	8	10	12	20	24	40	60	80	120	240
Irr. complex chars.	8	394	0	0	0	0	0	0	0	0	0	0	0	0	402
Irr. rational chars.	4	4	2	2	1	2	1	3	1	2	2	1	1	1	27

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$792$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	240
Arbitrary	2	4	20

Constructions

Presentation:	$\langle a, b \mid a^{2}=b^{792}=1, b^{a}=b^{197} \rangle$
Permutation group:	Degree $28$ $\langle(2,4)(3,7)(5,8)(6,9)(11,14)(15,17)(19,20)(21,22)(23,24)(25,26)(27,28), (10,11,12,15,13,14,16,17) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 196 \end{array}\right), \left(\begin{array}{rr} 27 & 144 \\ 72 & 27 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{197})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{88}$ $\,\rtimes\,$ $D_9$	$C_8$ $\,\rtimes\,$ $D_{99}$	$C_{72}$ $\,\rtimes\,$ $D_{11}$	$C_{792}$ $\,\rtimes\,$ $C_2$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{264}$ . $S_3$	$D_{198}$ . $C_4$	$C_{132}$ . $D_6$	$C_{44}$ . $D_{18}$	all 19

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

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 1116 subgroups in 60 conjugacy classes, 29 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $D_{198}$
Commutator:	$G' \simeq$ $C_{198}$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_{12}$	$G/\Phi \simeq$ $D_{66}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{792}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{792}:C_2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{66}$	$G/\operatorname{soc} \simeq$ $C_4\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

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 1584: $C_{792}:C_2$

Order 792: $C_{792}$, $C_{99}:C_8$, $C_4\times D_{99}$

Order 528: $C_{264}:C_2$

Order 396: $D_{198}$, $C_{396}$, $C_{99}:C_4$

Order 264: $C_{264}$, $C_{33}:C_8$, $C_4\times D_{33}$

Order 198: $C_{198}$, $D_{99}$

Order 176: $C_{88}:C_2$

Order 144: $C_{72}:C_2$

Order 132: $D_{66}$, $C_{132}$, $C_{33}:C_4$

Order 99: $C_{99}$

Order 88: $C_4\times D_{11}$, $C_{88}$, $C_{11}:C_8$

Order 72: $C_4\times D_9$, $C_{72}$, $C_9:C_8$

Order 66: $C_{66}$, $D_{33}$

Order 48: $C_{24}:C_2$

Order 44: $D_{22}$, $C_{44}$, $C_{11}:C_4$

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

Order 33: $C_{33}$

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

Order 22: $C_{22}$, $D_{11}$

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

Order 16: $\OD_{16}$

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

Order 11: $C_{11}$

Order 9: $C_9$

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 1584: $C_{792}:C_2$

Order 792: $C_{792}$, $C_{99}:C_8$, $C_4\times D_{99}$

Order 528: $C_{264}:C_2$

Order 396: $D_{198}$, $C_{396}$, $C_{99}:C_4$

Order 264: $C_{264}$, $C_{33}:C_8$, $C_4\times D_{33}$

Order 198: $C_{198}$, $D_{99}$

Order 176: $C_{88}:C_2$

Order 144: $C_{72}:C_2$

Order 132: $D_{66}$, $C_{132}$, $C_{33}:C_4$

Order 99: $C_{99}$

Order 88: $C_4\times D_{11}$, $C_{88}$, $C_{11}:C_8$

Order 72: $C_4\times D_9$, $C_{72}$, $C_9:C_8$

Order 66: $C_{66}$, $D_{33}$

Order 48: $C_{24}:C_2$

Order 44: $D_{22}$, $C_{44}$, $C_{11}:C_4$

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

Order 33: $C_{33}$

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

Order 22: $C_{22}$, $D_{11}$

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

Order 16: $\OD_{16}$

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

Order 11: $C_{11}$

Order 9: $C_9$

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 1584: $C_{792}:C_2$ ($C_1$)

Order 792: $C_{792}$ ($C_2$), $C_{99}:C_8$ ($C_2$), $C_4\times D_{99}$ ($C_2$)

Order 396: $D_{198}$ ($C_4$), $C_{396}$ ($C_2^2$), $C_{99}:C_4$ ($C_4$)

Order 264: $C_{264}$ ($S_3$)

Order 198: $C_{198}$ ($C_2\times C_4$)

Order 132: $C_{132}$ ($D_6$)

Order 99: $C_{99}$ ($\OD_{16}$)

Order 88: $C_{88}$ ($D_9$)

Order 72: $C_{72}$ ($D_{11}$)

Order 66: $C_{66}$ ($C_4\times S_3$)

Order 44: $C_{44}$ ($D_{18}$)

Order 36: $C_{36}$ ($D_{22}$)

Order 33: $C_{33}$ ($C_{24}:C_2$)

Order 24: $C_{24}$ ($D_{33}$)

Order 22: $C_{22}$ ($C_4\times D_9$)

Order 18: $C_{18}$ ($C_4\times D_{11}$)

Order 12: $C_{12}$ ($D_{66}$)

Order 11: $C_{11}$ ($C_{72}:C_2$)

Order 9: $C_9$ ($C_{88}:C_2$)

Order 8: $C_8$ ($D_{99}$)

Order 6: $C_6$ ($C_4\times D_{33}$)

Order 4: $C_4$ ($D_{198}$)

Order 3: $C_3$ ($C_{264}:C_2$)

Order 2: $C_2$ ($C_4\times D_{99}$)

Order 1: $C_1$ ($C_{792}:C_2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1584: $C_{792}:C_2$ ($C_1$)

Order 792: $C_{792}$ ($C_2$), $C_{99}:C_8$ ($C_2$), $C_4\times D_{99}$ ($C_2$)

Order 396: $D_{198}$ ($C_4$), $C_{396}$ ($C_2^2$), $C_{99}:C_4$ ($C_4$)

Order 264: $C_{264}$ ($S_3$)

Order 198: $C_{198}$ ($C_2\times C_4$)

Order 132: $C_{132}$ ($D_6$)

Order 99: $C_{99}$ ($\OD_{16}$)

Order 88: $C_{88}$ ($D_9$)

Order 72: $C_{72}$ ($D_{11}$)

Order 66: $C_{66}$ ($C_4\times S_3$)

Order 44: $C_{44}$ ($D_{18}$)

Order 36: $C_{36}$ ($D_{22}$)

Order 33: $C_{33}$ ($C_{24}:C_2$)

Order 24: $C_{24}$ ($D_{33}$)

Order 22: $C_{22}$ ($C_4\times D_9$)

Order 18: $C_{18}$ ($C_4\times D_{11}$)

Order 12: $C_{12}$ ($D_{66}$)

Order 11: $C_{11}$ ($C_{72}:C_2$)

Order 9: $C_9$ ($C_{88}:C_2$)

Order 8: $C_8$ ($D_{99}$)

Order 6: $C_6$ ($C_4\times D_{33}$)

Order 4: $C_4$ ($D_{198}$)

Order 3: $C_3$ ($C_{264}:C_2$)

Order 2: $C_2$ ($C_4\times D_{99}$)

Order 1: $C_1$ ($C_{792}:C_2$)

Series

Derived series

$C_{792}:C_2$

$\rhd$

$C_{198}$

$\rhd$

$C_1$

Chief series

$C_{792}:C_2$

$\rhd$

$C_4\times D_{99}$

$\rhd$

$C_{396}$

$\rhd$

$C_{198}$

$\rhd$

$C_{99}$

$\rhd$

$C_{33}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{792}:C_2$

$\rhd$

$C_{198}$

$\rhd$

$C_{99}$

Upper central series

$C_1$

$\lhd$

$C_4$

$\lhd$

$C_8$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $402 \times 402$ 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 $27 \times 27$ rational character table.