Abstract group 4752.h: $D_{132}:C_{18}$

• Label: 4752.h
• Order: \( 2^{4} \cdot 3^{3} \cdot 11 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{2} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \cdot 5 \)
• Perm deg.: $31$
• Trans deg.: $792$
• Rank: $3$

Group information

Description:	$D_{132}:C_{18}$
Order:	\(4752\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 11 \)
Exponent:	\(396\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 11 \)
Automorphism group:	$C_{66}.C_{30}.C_2^5$, of order \(63360\)\(\medspace = 2^{7} \cdot 3^{2} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 4, $C_3$ x 3, $C_{11}$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	9	11	12	18	22	33	36	44	66	99	132	198	396
Elements	1	135	8	136	288	18	10	296	846	30	80	864	40	240	180	320	540	720	4752
Conjugacy classes	1	4	5	5	17	12	5	22	42	15	40	54	20	120	90	160	270	360	1242
Divisions	1	4	3	4	9	2	1	8	7	2	3	6	2	7	2	6	5	4	76
Autjugacy classes	1	3	3	3	7	2	1	7	5	2	3	5	2	6	2	6	4	4	66

Dimension	1	2	4	6	8	10	12	20	24	40	60	80	120	240
Irr. complex chars.	72	1170	0	0	0	0	0	0	0	0	0	0	0	0	1242
Irr. rational chars.	8	12	5	8	4	4	4	8	3	5	4	4	4	3	76

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$792$
Rank:	$3$
Inequivalent generating triples:	$471744$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	2	not computed	240
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{198}=c^{12}=[b,c]=1, b^{a}=b^{109}c^{6}, c^{a}=c^{11} \rangle$
Permutation group:	Degree $31$ $\langle(1,2)(3,5)(4,7)(6,8)(9,10,11,12,13,14,15,16,17)(18,19,21,25,24,28,27,20,23,26,22) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 25 & 0 \\ 0 & 321 \end{array}\right), \left(\begin{array}{rr} 125 & 0 \\ 0 & 54 \end{array}\right), \left(\begin{array}{rr} 0 & 1 \\ 1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{397})$
Direct product:	$C_9$ $\, \times\, $ $(D_{132}:C_2)$
Semidirect product:	$D_{132}$ $\,\rtimes\,$ $C_{18}$	$(C_2\times C_{396})$ $\,\rtimes\,$ $S_3$	$(C_{33}:Q_8)$ $\,\rtimes\,$ $C_{18}$	$(D_{66}:C_{18})$ $\,\rtimes\,$ $C_2$ (2)	all 22
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{396}$ . $D_6$ (2)	$C_{36}$ . $D_{66}$ (2)	$(D_{66}:C_6)$ . $C_6$ (2)	$C_{66}$ . $(C_6\times D_6)$	all 55

Elements of the group are displayed as matrices in $\GL_{2}(\F_{397})$.

Homology

Abelianization:	$C_{2}^{2} \times C_{18} \simeq C_{2}^{3} \times C_{9}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 2320 subgroups in 272 conjugacy classes, 105 normal (81 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{36}$	$G/Z \simeq$ $D_{66}$
Commutator:	$G' \simeq$ $C_{66}$	$G/G' \simeq$ $C_2^2\times C_{18}$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_6\times D_{66}$
Fitting:	$\operatorname{Fit} \simeq$ $C_6\times C_{396}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $D_{132}:C_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_{66}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 4752: $D_{132}:C_{18}$

Order 2376: $C_{36}\times D_{33}$ x 2, $D_{66}:C_{18}$ x 2, $C_6\times C_{396}$, $C_{132}.C_{18}$, $C_9\times D_{132}$

Order 1584: $D_{132}:C_6$, $D_{44}:C_{18}$

Order 1188: $C_9\times D_{66}$ x 2, $C_3\times C_{396}$ x 2, $C_{33}:C_{36}$ x 2, $C_6\times C_{198}$

Order 792: $D_{66}:C_6$ x 2, $C_{12}\times D_{33}$ x 2, $C_2\times C_{396}$ x 2, $C_{99}:D_4$ x 2, $D_{11}\times C_{36}$ x 2, $C_3\times D_{132}$, $C_{132}.C_6$, $C_9\times D_{44}$, $C_{99}:Q_8$, $C_6\times C_{132}$

Order 594: $C_3\times C_{198}$ x 2, $C_9\times D_{33}$ x 2

Order 528: $D_{132}:C_2$, $D_{44}:C_6$

Order 432: $D_{12}:C_{18}$

Order 396: $C_{396}$ x 4, $C_9\times D_{22}$ x 2, $C_3\times D_{66}$ x 2, $C_{11}:C_{36}$ x 2, $C_3\times C_{132}$ x 2, $C_{33}:C_{12}$ x 2, $C_2\times C_{198}$ x 2, $C_6\times C_{66}$

Order 297: $C_3\times C_{99}$

Order 264: $C_2\times C_{132}$ x 3, $C_{33}:D_4$ x 2, $C_4\times D_{33}$ x 2, $C_{33}:D_4$ x 2, $C_{12}\times D_{11}$ x 2, $D_{132}$, $C_{33}:Q_8$, $C_3\times D_{44}$, $C_{33}:Q_8$

Order 216: $D_6:C_{18}$ x 2, $S_3\times C_{36}$ x 2, $C_6\times C_{36}$, $C_9\times D_{12}$, $C_{12}.C_{18}$

Order 198: $C_{198}$ x 5, $C_3\times D_{33}$ x 2, $C_9\times D_{11}$ x 2, $C_3\times C_{66}$ x 2

Order 176: $D_{44}:C_2$

Order 144: $D_4:C_{18}$, $D_{12}:C_6$

Order 132: $C_{132}$ x 6, $C_2\times C_{66}$ x 3, $D_{66}$ x 2, $C_3\times D_{22}$ x 2, $C_{33}:C_4$ x 2, $C_{11}:C_{12}$ x 2

Order 108: $C_3:C_{36}$ x 2, $S_3\times C_{18}$ x 2, $C_3\times C_{36}$ x 2, $C_6\times C_{18}$

Order 99: $C_{99}$ x 2, $C_3\times C_{33}$

Order 88: $C_{11}:D_4$ x 2, $C_4\times D_{11}$ x 2, $C_2\times C_{44}$, $D_{44}$, $C_{11}:Q_8$

Order 72: $C_2\times C_{36}$ x 4, $D_4\times C_9$ x 3, $C_6\wr C_2$ x 2, $S_3\times C_{12}$ x 2, $C_6\times C_{12}$, $C_3\times D_{12}$, $C_3^2:Q_8$, $Q_8\times C_9$

Order 66: $C_{66}$ x 7, $D_{33}$ x 2, $C_3\times D_{11}$ x 2

Order 54: $C_3\times C_{18}$ x 2, $S_3\times C_9$ x 2

Order 48: $D_4:C_6$, $D_{12}:C_2$

Order 44: $D_{22}$ x 2, $C_{44}$ x 2, $C_{11}:C_4$ x 2, $C_2\times C_{22}$

Order 36: $C_{36}$ x 6, $C_2\times C_{18}$ x 4, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_6^2$

Order 33: $C_{33}$ x 3

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 5, $C_3\times D_4$ x 3, $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $D_{12}$, $C_3:Q_8$, $C_3\times Q_8$

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

Order 18: $C_{18}$ x 7, $C_3\times C_6$ x 2, $C_3\times S_3$ x 2

Order 16: $D_4:C_2$

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 5, $D_6$ x 2, $C_3:C_4$ x 2

Order 11: $C_{11}$

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$

Order 6: $C_6$ x 9, $S_3$ x 2

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4752: $D_{132}:C_{18}$

Order 2376: $C_{36}\times D_{33}$, $C_6\times C_{396}$, $C_{132}.C_{18}$, $C_9\times D_{132}$, $D_{66}:C_{18}$

Order 1584: $D_{132}:C_6$, $D_{44}:C_{18}$

Order 1188: $C_3\times C_{396}$ x 2, $C_6\times C_{198}$, $C_9\times D_{66}$, $C_{33}:C_{36}$

Order 792: $C_2\times C_{396}$ x 2, $D_{66}:C_6$, $C_3\times D_{132}$, $C_{12}\times D_{33}$, $C_{132}.C_6$, $C_{99}:D_4$, $C_9\times D_{44}$, $D_{11}\times C_{36}$, $C_{99}:Q_8$, $C_6\times C_{132}$

Order 594: $C_3\times C_{198}$ x 2, $C_9\times D_{33}$

Order 528: $D_{132}:C_2$, $D_{44}:C_6$

Order 432: $D_{12}:C_{18}$

Order 396: $C_{396}$ x 4, $C_3\times C_{132}$ x 2, $C_2\times C_{198}$ x 2, $C_9\times D_{22}$, $C_6\times C_{66}$, $C_3\times D_{66}$, $C_{11}:C_{36}$, $C_{33}:C_{12}$

Order 297: $C_3\times C_{99}$

Order 264: $C_2\times C_{132}$ x 3, $C_{33}:D_4$, $D_{132}$, $C_4\times D_{33}$, $C_{33}:Q_8$, $C_{33}:D_4$, $C_3\times D_{44}$, $C_{12}\times D_{11}$, $C_{33}:Q_8$

Order 216: $C_6\times C_{36}$, $D_6:C_{18}$, $C_9\times D_{12}$, $S_3\times C_{36}$, $C_{12}.C_{18}$

Order 198: $C_{198}$ x 4, $C_3\times C_{66}$ x 2, $C_3\times D_{33}$, $C_9\times D_{11}$

Order 176: $D_{44}:C_2$

Order 144: $D_4:C_{18}$, $D_{12}:C_6$

Order 132: $C_{132}$ x 6, $C_2\times C_{66}$ x 3, $D_{66}$, $C_3\times D_{22}$, $C_{33}:C_4$, $C_{11}:C_{12}$

Order 108: $C_3\times C_{36}$ x 2, $C_3:C_{36}$, $C_6\times C_{18}$, $S_3\times C_{18}$

Order 99: $C_{99}$ x 2, $C_3\times C_{33}$

Order 88: $C_2\times C_{44}$, $C_{11}:D_4$, $D_{44}$, $C_4\times D_{11}$, $C_{11}:Q_8$

Order 72: $C_2\times C_{36}$ x 3, $D_4\times C_9$ x 2, $C_6\times C_{12}$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $Q_8\times C_9$

Order 66: $C_{66}$ x 6, $D_{33}$, $C_3\times D_{11}$

Order 54: $C_3\times C_{18}$ x 2, $S_3\times C_9$

Order 48: $D_4:C_6$, $D_{12}:C_2$

Order 44: $C_{44}$ x 2, $C_2\times C_{22}$, $D_{22}$, $C_{11}:C_4$

Order 36: $C_{36}$ x 5, $C_2\times C_{18}$ x 3, $C_3\times C_{12}$ x 2, $C_3:C_{12}$, $C_6^2$, $C_6\times S_3$

Order 33: $C_{33}$ x 3

Order 27: $C_3\times C_9$

Order 24: $C_2\times C_{12}$ x 4, $C_3\times D_4$ x 2, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_3\times Q_8$

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

Order 18: $C_{18}$ x 5, $C_3\times C_6$ x 2, $C_3\times S_3$

Order 16: $D_4:C_2$

Order 12: $C_{12}$ x 7, $C_2\times C_6$ x 4, $D_6$, $C_3:C_4$

Order 11: $C_{11}$

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $D_4$ x 2, $C_2\times C_4$ x 2, $Q_8$

Order 6: $C_6$ x 7, $S_3$

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4752: $D_{132}:C_{18}$ ($C_1$)

Order 2376: $C_{36}\times D_{33}$ ($C_2$) x 2, $D_{66}:C_{18}$ ($C_2$) x 2, $C_6\times C_{396}$ ($C_2$), $C_{132}.C_{18}$ ($C_2$), $C_9\times D_{132}$ ($C_2$)

Order 1584: $D_{132}:C_6$ ($C_3$)

Order 1188: $C_9\times D_{66}$ ($C_2^2$) x 2, $C_3\times C_{396}$ ($C_2^2$) x 2, $C_{33}:C_{36}$ ($C_2^2$) x 2, $C_6\times C_{198}$ ($C_2^2$)

Order 792: $D_{66}:C_6$ ($C_6$) x 2, $C_{12}\times D_{33}$ ($C_6$) x 2, $C_3\times D_{132}$ ($C_6$), $C_{132}.C_6$ ($C_6$), $C_2\times C_{396}$ ($S_3$), $C_6\times C_{132}$ ($C_6$)

Order 594: $C_3\times C_{198}$ ($C_2^3$)

Order 528: $D_{132}:C_2$ ($C_9$)

Order 396: $C_{396}$ ($D_6$) x 2, $C_3\times D_{66}$ ($C_2\times C_6$) x 2, $C_3\times C_{132}$ ($C_2\times C_6$) x 2, $C_{33}:C_{12}$ ($C_2\times C_6$) x 2, $C_6\times C_{66}$ ($C_2\times C_6$), $C_2\times C_{198}$ ($D_6$)

Order 297: $C_3\times C_{99}$ ($D_4:C_2$)

Order 264: $C_{33}:D_4$ ($C_{18}$) x 2, $C_4\times D_{33}$ ($C_{18}$) x 2, $C_2\times C_{132}$ ($C_3\times S_3$), $C_2\times C_{132}$ ($C_{18}$), $D_{132}$ ($C_{18}$), $C_{33}:Q_8$ ($C_{18}$)

Order 216: $C_6\times C_{36}$ ($D_{11}$)

Order 198: $C_{198}$ ($C_2\times D_6$), $C_3\times C_{66}$ ($C_2^2\times C_6$)

Order 132: $D_{66}$ ($C_2\times C_{18}$) x 2, $C_{132}$ ($C_2\times C_{18}$) x 2, $C_{132}$ ($C_6\times S_3$) x 2, $C_{33}:C_4$ ($C_2\times C_{18}$) x 2, $C_2\times C_{66}$ ($C_2\times C_{18}$), $C_2\times C_{66}$ ($C_6\times S_3$)

Order 108: $C_3\times C_{36}$ ($D_{22}$) x 2, $C_6\times C_{18}$ ($D_{22}$)

Order 99: $C_3\times C_{33}$ ($D_4:C_6$), $C_{99}$ ($D_{12}:C_2$)

Order 88: $C_2\times C_{44}$ ($S_3\times C_9$)

Order 72: $C_2\times C_{36}$ ($D_{33}$), $C_6\times C_{12}$ ($C_3\times D_{11}$)

Order 66: $C_{66}$ ($C_6\times D_6$), $C_{66}$ ($C_2^2\times C_{18}$)

Order 54: $C_3\times C_{18}$ ($C_2\times D_{22}$)

Order 44: $C_{44}$ ($S_3\times C_{18}$) x 2, $C_2\times C_{22}$ ($S_3\times C_{18}$)

Order 36: $C_3\times C_{12}$ ($C_3\times D_{22}$) x 2, $C_{36}$ ($D_{66}$) x 2, $C_2\times C_{18}$ ($D_{66}$), $C_6^2$ ($C_3\times D_{22}$)

Order 33: $C_{33}$ ($D_4:C_{18}$), $C_{33}$ ($D_{12}:C_6$)

Order 27: $C_3\times C_9$ ($D_{44}:C_2$)

Order 24: $C_2\times C_{12}$ ($C_3\times D_{33}$), $C_2\times C_{12}$ ($C_9\times D_{11}$)

Order 22: $C_{22}$ ($D_6\times C_{18}$)

Order 18: $C_3\times C_6$ ($C_6\times D_{22}$), $C_{18}$ ($C_2\times D_{66}$)

Order 12: $C_{12}$ ($C_9\times D_{22}$) x 2, $C_{12}$ ($C_3\times D_{66}$) x 2, $C_2\times C_6$ ($C_9\times D_{22}$), $C_2\times C_6$ ($C_3\times D_{66}$)

Order 11: $C_{11}$ ($D_{12}:C_{18}$)

Order 9: $C_3^2$ ($D_{44}:C_6$), $C_9$ ($D_{132}:C_2$)

Order 8: $C_2\times C_4$ ($C_9\times D_{33}$)

Order 6: $C_6$ ($C_{18}\times D_{22}$), $C_6$ ($C_6\times D_{66}$)

Order 4: $C_4$ ($C_9\times D_{66}$) x 2, $C_2^2$ ($C_9\times D_{66}$)

Order 3: $C_3$ ($D_{132}:C_6$), $C_3$ ($D_{44}:C_{18}$)

Order 2: $C_2$ ($C_{11}:(C_2\times C_{18}\times S_3)$)

Order 1: $C_1$ ($D_{132}:C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4752: $D_{132}:C_{18}$ ($C_1$)

Order 2376: $C_{36}\times D_{33}$ ($C_2$), $C_6\times C_{396}$ ($C_2$), $C_{132}.C_{18}$ ($C_2$), $C_9\times D_{132}$ ($C_2$), $D_{66}:C_{18}$ ($C_2$)

Order 1584: $D_{132}:C_6$ ($C_3$)

Order 1188: $C_3\times C_{396}$ ($C_2^2$) x 2, $C_6\times C_{198}$ ($C_2^2$), $C_9\times D_{66}$ ($C_2^2$), $C_{33}:C_{36}$ ($C_2^2$)

Order 792: $D_{66}:C_6$ ($C_6$), $C_3\times D_{132}$ ($C_6$), $C_{12}\times D_{33}$ ($C_6$), $C_{132}.C_6$ ($C_6$), $C_2\times C_{396}$ ($S_3$), $C_6\times C_{132}$ ($C_6$)

Order 594: $C_3\times C_{198}$ ($C_2^3$)

Order 528: $D_{132}:C_2$ ($C_9$)

Order 396: $C_{396}$ ($D_6$) x 2, $C_3\times C_{132}$ ($C_2\times C_6$) x 2, $C_6\times C_{66}$ ($C_2\times C_6$), $C_3\times D_{66}$ ($C_2\times C_6$), $C_{33}:C_{12}$ ($C_2\times C_6$), $C_2\times C_{198}$ ($D_6$)

Order 297: $C_3\times C_{99}$ ($D_4:C_2$)

Order 264: $C_2\times C_{132}$ ($C_3\times S_3$), $C_2\times C_{132}$ ($C_{18}$), $C_{33}:D_4$ ($C_{18}$), $D_{132}$ ($C_{18}$), $C_4\times D_{33}$ ($C_{18}$), $C_{33}:Q_8$ ($C_{18}$)

Order 216: $C_6\times C_{36}$ ($D_{11}$)

Order 198: $C_{198}$ ($C_2\times D_6$), $C_3\times C_{66}$ ($C_2^2\times C_6$)

Order 132: $C_{132}$ ($C_2\times C_{18}$) x 2, $C_{132}$ ($C_6\times S_3$) x 2, $D_{66}$ ($C_2\times C_{18}$), $C_{33}:C_4$ ($C_2\times C_{18}$), $C_2\times C_{66}$ ($C_2\times C_{18}$), $C_2\times C_{66}$ ($C_6\times S_3$)

Order 108: $C_3\times C_{36}$ ($D_{22}$) x 2, $C_6\times C_{18}$ ($D_{22}$)

Order 99: $C_3\times C_{33}$ ($D_4:C_6$), $C_{99}$ ($D_{12}:C_2$)

Order 88: $C_2\times C_{44}$ ($S_3\times C_9$)

Order 72: $C_2\times C_{36}$ ($D_{33}$), $C_6\times C_{12}$ ($C_3\times D_{11}$)

Order 66: $C_{66}$ ($C_6\times D_6$), $C_{66}$ ($C_2^2\times C_{18}$)

Order 54: $C_3\times C_{18}$ ($C_2\times D_{22}$)

Order 44: $C_{44}$ ($S_3\times C_{18}$) x 2, $C_2\times C_{22}$ ($S_3\times C_{18}$)

Order 36: $C_3\times C_{12}$ ($C_3\times D_{22}$) x 2, $C_{36}$ ($D_{66}$) x 2, $C_2\times C_{18}$ ($D_{66}$), $C_6^2$ ($C_3\times D_{22}$)

Order 33: $C_{33}$ ($D_4:C_{18}$), $C_{33}$ ($D_{12}:C_6$)

Order 27: $C_3\times C_9$ ($D_{44}:C_2$)

Order 24: $C_2\times C_{12}$ ($C_3\times D_{33}$), $C_2\times C_{12}$ ($C_9\times D_{11}$)

Order 22: $C_{22}$ ($D_6\times C_{18}$)

Order 18: $C_3\times C_6$ ($C_6\times D_{22}$), $C_{18}$ ($C_2\times D_{66}$)

Order 12: $C_{12}$ ($C_9\times D_{22}$) x 2, $C_{12}$ ($C_3\times D_{66}$) x 2, $C_2\times C_6$ ($C_9\times D_{22}$), $C_2\times C_6$ ($C_3\times D_{66}$)

Order 11: $C_{11}$ ($D_{12}:C_{18}$)

Order 9: $C_3^2$ ($D_{44}:C_6$), $C_9$ ($D_{132}:C_2$)

Order 8: $C_2\times C_4$ ($C_9\times D_{33}$)

Order 6: $C_6$ ($C_{18}\times D_{22}$), $C_6$ ($C_6\times D_{66}$)

Order 4: $C_4$ ($C_9\times D_{66}$) x 2, $C_2^2$ ($C_9\times D_{66}$)

Order 3: $C_3$ ($D_{132}:C_6$), $C_3$ ($D_{44}:C_{18}$)

Order 2: $C_2$ ($C_{11}:(C_2\times C_{18}\times S_3)$)

Order 1: $C_1$ ($D_{132}:C_{18}$)

Series

Derived series

$D_{132}:C_{18}$

$\rhd$

$C_{66}$

$\rhd$

$C_1$

Chief series

$D_{132}:C_{18}$

$\rhd$

$C_6\times C_{396}$

$\rhd$

$C_6\times C_{198}$

$\rhd$

$C_3\times C_{198}$

$\rhd$

$C_3\times C_{66}$

$\rhd$

$C_{66}$

$\rhd$

$C_{33}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$D_{132}:C_{18}$

$\rhd$

$C_{66}$

$\rhd$

$C_{33}$

Upper central series

$C_1$

$\lhd$

$C_{36}$

$\lhd$

$C_2\times C_{36}$

Supergroups

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

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

The $1242 \times 1242$ character table is not available for this group.

Rational character table

See the $76 \times 76$ rational character table.