Abstract group 792.17: $C_{99}:D_4$

• Label: 792.17
• Order: \( 2^{3} \cdot 3^{2} \cdot 11 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \cdot 5 \)
• Perm deg.: $24$
• Trans deg.: $396$
• Rank: $2$

Group information

Description:	$C_{99}:D_4$
Order:	\(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \)
Exponent:	\(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \)
Automorphism group:	$C_2\times C_{11}:(C_2^2\times C_{30})$, of order \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 3, $C_3$ x 2, $C_{11}$
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	9	11	12	18	22	33	36	66	99	198
Elements	1	25	2	22	50	6	10	44	150	30	20	132	60	60	180	792
Conjugacy classes	1	3	2	1	6	6	5	2	18	15	10	6	30	30	90	225
Divisions	1	3	1	1	3	1	1	1	3	2	1	1	2	1	2	24
Autjugacy classes	1	3	1	1	3	1	1	1	3	2	1	1	2	1	2	24

Dimension	1	2	4	6	10	12	20	40	60	120
Irr. complex chars.	36	189	0	0	0	0	0	0	0	0	225
Irr. rational chars.	4	5	1	4	2	1	3	1	2	1	24

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$396$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	4	120
Arbitrary	2	4	18

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{198}=[a,b]=[b,c]=1, c^{a}=bc^{109} \rangle$
Permutation group:	Degree $24$ $\langle(11,13)(15,16)(17,18)(19,20)(21,22)(23,24), (10,11)(12,13), (1,2,4,3,5,7,6,8,9) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 181 & 0 \\ 0 & 11 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 145 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$
Direct product:	$C_9$ $\, \times\, $ $(C_{11}:D_4)$
Semidirect product:	$C_{99}$ $\,\rtimes\,$ $D_4$	$D_{22}$ $\,\rtimes\,$ $C_{18}$	$(C_9\times D_{22})$ $\,\rtimes\,$ $C_2$	$C_{11}$ $\,\rtimes\,$ $(D_4\times C_9)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{18}$ . $D_{22}$	$C_{198}$ . $C_2^2$	$(C_{33}:D_4)$ . $C_3$	$(C_3\times D_{22})$ . $C_6$	all 13

Elements of the group are displayed as matrices in $\GL_{2}(\F_{199})$.

Homology

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

Subgroups

There are 210 subgroups in 48 conjugacy classes, 27 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{18}$	$G/Z \simeq$ $D_{22}$
Commutator:	$G' \simeq$ $C_{22}$	$G/G' \simeq$ $C_2\times C_{18}$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_3\times D_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{198}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{99}:D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{66}$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
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 792: $C_{99}:D_4$

Order 396: $C_9\times D_{22}$, $C_{11}:C_{36}$, $C_2\times C_{198}$

Order 264: $C_{33}:D_4$

Order 198: $C_{198}$ x 2, $C_9\times D_{11}$

Order 132: $C_3\times D_{22}$, $C_{11}:C_{12}$, $C_2\times C_{66}$

Order 99: $C_{99}$

Order 88: $C_{11}:D_4$

Order 72: $D_4\times C_9$

Order 66: $C_{66}$ x 2, $C_3\times D_{11}$

Order 44: $C_2\times C_{22}$, $D_{22}$, $C_{11}:C_4$

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

Order 33: $C_{33}$

Order 24: $C_3\times D_4$

Order 22: $C_{22}$ x 2, $D_{11}$

Order 18: $C_{18}$ x 3

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

Order 11: $C_{11}$

Order 9: $C_9$

Order 8: $D_4$

Order 6: $C_6$ x 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 792: $C_{99}:D_4$

Order 396: $C_9\times D_{22}$, $C_{11}:C_{36}$, $C_2\times C_{198}$

Order 264: $C_{33}:D_4$

Order 198: $C_{198}$ x 2, $C_9\times D_{11}$

Order 132: $C_3\times D_{22}$, $C_{11}:C_{12}$, $C_2\times C_{66}$

Order 99: $C_{99}$

Order 88: $C_{11}:D_4$

Order 72: $D_4\times C_9$

Order 66: $C_{66}$ x 2, $C_3\times D_{11}$

Order 44: $C_2\times C_{22}$, $D_{22}$, $C_{11}:C_4$

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

Order 33: $C_{33}$

Order 24: $C_3\times D_4$

Order 22: $C_{22}$ x 2, $D_{11}$

Order 18: $C_{18}$ x 3

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

Order 11: $C_{11}$

Order 9: $C_9$

Order 8: $D_4$

Order 6: $C_6$ x 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 792: $C_{99}:D_4$ ($C_1$)

Order 396: $C_9\times D_{22}$ ($C_2$), $C_{11}:C_{36}$ ($C_2$), $C_2\times C_{198}$ ($C_2$)

Order 264: $C_{33}:D_4$ ($C_3$)

Order 198: $C_{198}$ ($C_2^2$)

Order 132: $C_3\times D_{22}$ ($C_6$), $C_{11}:C_{12}$ ($C_6$), $C_2\times C_{66}$ ($C_6$)

Order 99: $C_{99}$ ($D_4$)

Order 88: $C_{11}:D_4$ ($C_9$)

Order 66: $C_{66}$ ($C_2\times C_6$)

Order 44: $C_2\times C_{22}$ ($C_{18}$), $D_{22}$ ($C_{18}$), $C_{11}:C_4$ ($C_{18}$)

Order 36: $C_2\times C_{18}$ ($D_{11}$)

Order 33: $C_{33}$ ($C_3\times D_4$)

Order 22: $C_{22}$ ($C_2\times C_{18}$)

Order 18: $C_{18}$ ($D_{22}$)

Order 12: $C_2\times C_6$ ($C_3\times D_{11}$)

Order 11: $C_{11}$ ($D_4\times C_9$)

Order 9: $C_9$ ($C_{11}:D_4$)

Order 6: $C_6$ ($C_3\times D_{22}$)

Order 4: $C_2^2$ ($C_9\times D_{11}$)

Order 3: $C_3$ ($C_{33}:D_4$)

Order 2: $C_2$ ($C_9\times D_{22}$)

Order 1: $C_1$ ($C_{99}:D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 792: $C_{99}:D_4$ ($C_1$)

Order 396: $C_9\times D_{22}$ ($C_2$), $C_{11}:C_{36}$ ($C_2$), $C_2\times C_{198}$ ($C_2$)

Order 264: $C_{33}:D_4$ ($C_3$)

Order 198: $C_{198}$ ($C_2^2$)

Order 132: $C_3\times D_{22}$ ($C_6$), $C_{11}:C_{12}$ ($C_6$), $C_2\times C_{66}$ ($C_6$)

Order 99: $C_{99}$ ($D_4$)

Order 88: $C_{11}:D_4$ ($C_9$)

Order 66: $C_{66}$ ($C_2\times C_6$)

Order 44: $C_2\times C_{22}$ ($C_{18}$), $D_{22}$ ($C_{18}$), $C_{11}:C_4$ ($C_{18}$)

Order 36: $C_2\times C_{18}$ ($D_{11}$)

Order 33: $C_{33}$ ($C_3\times D_4$)

Order 22: $C_{22}$ ($C_2\times C_{18}$)

Order 18: $C_{18}$ ($D_{22}$)

Order 12: $C_2\times C_6$ ($C_3\times D_{11}$)

Order 11: $C_{11}$ ($D_4\times C_9$)

Order 9: $C_9$ ($C_{11}:D_4$)

Order 6: $C_6$ ($C_3\times D_{22}$)

Order 4: $C_2^2$ ($C_9\times D_{11}$)

Order 3: $C_3$ ($C_{33}:D_4$)

Order 2: $C_2$ ($C_9\times D_{22}$)

Order 1: $C_1$ ($C_{99}:D_4$)

Series

Derived series

$C_{99}:D_4$

$\rhd$

$C_{22}$

$\rhd$

$C_1$

Chief series

$C_{99}:D_4$

$\rhd$

$C_2\times C_{198}$

$\rhd$

$C_{198}$

$\rhd$

$C_{66}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{99}:D_4$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_{18}$

$\lhd$

$C_2\times C_{18}$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 10 larger groups in the database.

Character theory

Complex character table

See the $225 \times 225$ 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 $24 \times 24$ rational character table.