Abstract group 792.40: $D_9\times D_{22}$

• Label: 792.40
• Order: \( 2^{3} \cdot 3^{2} \cdot 11 \)
• Exponent: \( 2 \cdot 3^{2} \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{3} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \cdot 5 \)
• Perm deg.: $22$
• Trans deg.: $198$
• Rank: $3$

Group information

Description:	$D_9\times D_{22}$
Order:	\(792\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 11 \)
Exponent:	\(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \)
Automorphism group:	$C_{99}.C_{30}.C_2^3$, of order \(23760\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \cdot 11 \)
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 $p = 2$, metabelian, and an A-group.

Group statistics

Order	1	2	3	6	9	11	18	22	33	66	99	198
Elements	1	239	2	46	6	10	138	190	20	20	60	60	792
Conjugacy classes	1	7	1	3	3	5	9	15	5	5	15	15	84
Divisions	1	7	1	3	1	1	3	3	1	1	1	1	24
Autjugacy classes	1	4	1	2	1	1	2	2	1	1	1	1	18

Dimension	1	2	4	6	10	20	60
Irr. complex chars.	8	36	40	0	0	0	0	84
Irr. rational chars.	8	4	0	4	4	2	2	24

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$198$
Rank:	$3$
Inequivalent generating triples:	$6048$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	4	60
Arbitrary	4	4	16

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{198}=[a,b]=1, c^{a}=c^{109}, c^{b}=c^{89} \rangle$
Permutation group:	Degree $22$ $\langle(13,14)(15,16)(17,18)(19,20)(21,22), (2,4)(3,7)(5,8)(6,9)(10,11), (10,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 89 & 0 \\ 0 & 89 \end{array}\right), \left(\begin{array}{rr} 1 & 33 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 76 & 55 \\ 66 & 32 \end{array}\right), \left(\begin{array}{rr} 89 & 54 \\ 0 & 98 \end{array}\right), \left(\begin{array}{rr} 1 & 44 \\ 33 & 67 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/99\Z)$
Direct product:	$C_2$ $\, \times\, $ $D_9$ $\, \times\, $ $D_{11}$
Semidirect product:	$D_{198}$ $\,\rtimes\,$ $C_2$	$C_{22}$ $\,\rtimes\,$ $D_{18}$	$C_{18}$ $\,\rtimes\,$ $D_{22}$	$D_{99}$ $\,\rtimes\,$ $C_2^2$ (2)	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{66}$ . $D_6$	$(C_3\times D_{22})$ . $S_3$	$C_6$ . $(S_3\times D_{11})$	$C_3$ . $(S_3\times D_{22})$	all 6

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1446 subgroups in 96 conjugacy classes, 35 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_9\times D_{11}$
Commutator:	$G' \simeq$ $C_{99}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $S_3\times D_{22}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{198}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $D_9\times D_{22}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{66}$	$G/\operatorname{soc} \simeq$ $D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 792: $D_9\times D_{22}$

Order 396: $D_9\times D_{11}$ x 4, $D_{198}$, $C_{11}\times D_{18}$, $C_9\times D_{22}$

Order 264: $S_3\times D_{22}$

Order 198: $D_{99}$ x 2, $C_9\times D_{11}$ x 2, $C_{11}\times D_9$ x 2, $C_{198}$

Order 132: $S_3\times D_{11}$ x 4, $D_{66}$, $S_3\times C_{22}$, $C_3\times D_{22}$

Order 99: $C_{99}$

Order 88: $C_2\times D_{22}$

Order 72: $C_2\times D_{18}$

Order 66: $D_{33}$ x 2, $C_3\times D_{11}$ x 2, $S_3\times C_{11}$ x 2, $C_{66}$

Order 44: $D_{22}$ x 6, $C_2\times C_{22}$

Order 36: $D_{18}$ x 6, $C_2\times C_{18}$

Order 33: $C_{33}$

Order 24: $C_2\times D_6$

Order 22: $D_{11}$ x 4, $C_{22}$ x 3

Order 18: $D_9$ x 4, $C_{18}$ x 3

Order 12: $D_6$ x 6, $C_2\times C_6$

Order 11: $C_{11}$

Order 9: $C_9$

Order 8: $C_2^3$

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 792: $D_9\times D_{22}$

Order 396: $D_{198}$, $C_{11}\times D_{18}$, $C_9\times D_{22}$, $D_9\times D_{11}$

Order 264: $S_3\times D_{22}$

Order 198: $C_{198}$, $D_{99}$, $C_9\times D_{11}$, $C_{11}\times D_9$

Order 132: $D_{66}$, $S_3\times C_{22}$, $C_3\times D_{22}$, $S_3\times D_{11}$

Order 99: $C_{99}$

Order 88: $C_2\times D_{22}$

Order 72: $C_2\times D_{18}$

Order 66: $C_{66}$, $D_{33}$, $C_3\times D_{11}$, $S_3\times C_{11}$

Order 44: $D_{22}$ x 3, $C_2\times C_{22}$

Order 36: $D_{18}$ x 3, $C_2\times C_{18}$

Order 33: $C_{33}$

Order 24: $C_2\times D_6$

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

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

Order 12: $D_6$ x 3, $C_2\times C_6$

Order 11: $C_{11}$

Order 9: $C_9$

Order 8: $C_2^3$

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

Order 4: $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 792: $D_9\times D_{22}$ ($C_1$)

Order 396: $D_9\times D_{11}$ ($C_2$) x 4, $D_{198}$ ($C_2$), $C_{11}\times D_{18}$ ($C_2$), $C_9\times D_{22}$ ($C_2$)

Order 198: $D_{99}$ ($C_2^2$) x 2, $C_9\times D_{11}$ ($C_2^2$) x 2, $C_{11}\times D_9$ ($C_2^2$) x 2, $C_{198}$ ($C_2^2$)

Order 132: $C_3\times D_{22}$ ($S_3$)

Order 99: $C_{99}$ ($C_2^3$)

Order 66: $C_3\times D_{11}$ ($D_6$) x 2, $C_{66}$ ($D_6$)

Order 44: $D_{22}$ ($D_9$)

Order 36: $D_{18}$ ($D_{11}$)

Order 33: $C_{33}$ ($C_2\times D_6$)

Order 22: $D_{11}$ ($D_{18}$) x 2, $C_{22}$ ($D_{18}$)

Order 18: $D_9$ ($D_{22}$) x 2, $C_{18}$ ($D_{22}$)

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

Order 9: $C_9$ ($C_2\times D_{22}$)

Order 6: $C_6$ ($S_3\times D_{11}$)

Order 3: $C_3$ ($S_3\times D_{22}$)

Order 2: $C_2$ ($D_9\times D_{11}$)

Order 1: $C_1$ ($D_9\times D_{22}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 792: $D_9\times D_{22}$ ($C_1$)

Order 396: $D_{198}$ ($C_2$), $C_{11}\times D_{18}$ ($C_2$), $C_9\times D_{22}$ ($C_2$), $D_9\times D_{11}$ ($C_2$)

Order 198: $C_{198}$ ($C_2^2$), $D_{99}$ ($C_2^2$), $C_9\times D_{11}$ ($C_2^2$), $C_{11}\times D_9$ ($C_2^2$)

Order 132: $C_3\times D_{22}$ ($S_3$)

Order 99: $C_{99}$ ($C_2^3$)

Order 66: $C_{66}$ ($D_6$), $C_3\times D_{11}$ ($D_6$)

Order 44: $D_{22}$ ($D_9$)

Order 36: $D_{18}$ ($D_{11}$)

Order 33: $C_{33}$ ($C_2\times D_6$)

Order 22: $C_{22}$ ($D_{18}$), $D_{11}$ ($D_{18}$)

Order 18: $C_{18}$ ($D_{22}$), $D_9$ ($D_{22}$)

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

Order 9: $C_9$ ($C_2\times D_{22}$)

Order 6: $C_6$ ($S_3\times D_{11}$)

Order 3: $C_3$ ($S_3\times D_{22}$)

Order 2: $C_2$ ($D_9\times D_{11}$)

Order 1: $C_1$ ($D_9\times D_{22}$)

Series

Derived series

$D_9\times D_{22}$

$\rhd$

$C_{99}$

$\rhd$

$C_1$

Chief series

$D_9\times D_{22}$

$\rhd$

$D_9\times D_{11}$

$\rhd$

$C_9\times D_{11}$

$\rhd$

$C_{99}$

$\rhd$

$C_{33}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$D_9\times D_{22}$

$\rhd$

$C_{99}$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

This group is a maximal quotient of 28 larger groups in the database.

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

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