Abstract group 1320.18: $D_6:F_{11}$

• Label: 1320.18
• Order: \( 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $18$
• Trans deg.: $132$
• Rank: $2$

Group information

Description:	$D_6:F_{11}$
Order:	\(1320\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Exponent:	\(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}:(C_2^2\times C_{10}\times S_3)$, of order \(2640\)\(\medspace = 2^{4} \cdot 3 \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 3, $C_3$, $C_5$, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	5	6	10	11	15	20	22	30	33	66
Elements	1	29	2	66	44	46	396	10	88	264	70	264	20	20	1320
Conjugacy classes	1	3	1	1	4	3	12	1	4	4	3	12	1	1	51
Divisions	1	3	1	1	1	2	3	1	1	1	2	2	1	1	21
Autjugacy classes	1	3	1	1	4	2	12	1	4	4	2	8	1	1	45

Dimension	1	2	4	8	10	16	20
Irr. complex chars.	20	25	0	0	4	0	2	51
Irr. rational chars.	4	3	5	3	2	1	3	21

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$132$
Rank:	$2$
Inequivalent generating pairs:	$144$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	20	40	40
Arbitrary	12	14	14

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{30}=c^{22}=[a,c]=1, b^{a}=b^{11}c^{11}, c^{b}=c^{13} \rangle$
Permutation group:	Degree $18$ $\langle(2,3)(4,5)(6,7)(8,10)(9,11)(13,15), (12,13)(14,15)(17,18), (2,4,6,8,9)(3,5,7,10,11), (12,14)(13,15), (16,17,18), (1,2,5,8,6,9,11,7,10,4,3)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 23 & 0 \\ 0 & 23 \end{array}\right), \left(\begin{array}{rr} 23 & 33 \\ 11 & 12 \end{array}\right), \left(\begin{array}{rr} 23 & 38 \\ 22 & 21 \end{array}\right), \left(\begin{array}{rr} 23 & 33 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/44\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$D_6$ $\,\rtimes\,$ $F_{11}$	$D_{22}$ $\,\rtimes\,$ $(C_5\times S_3)$	$(C_2\times F_{11})$ $\,\rtimes\,$ $S_3$	$(C_{33}:D_4)$ $\,\rtimes\,$ $C_5$	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(S_3\times F_{11})$	$C_6$ . $(C_2\times F_{11})$	$C_{22}$ . $(S_3\times C_{10})$	$(C_{11}:C_{10})$ . $D_6$	all 6

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

Homology

Abelianization:	$C_{2} \times C_{10} \simeq C_{2}^{2} \times C_{5}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 560 subgroups in 64 conjugacy classes, 23 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3\times F_{11}$
Commutator:	$G' \simeq$ $C_{66}$	$G/G' \simeq$ $C_2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $S_3\times F_{11}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{66}$	$G/\operatorname{Fit} \simeq$ $C_2\times C_{10}$
Radical:	$R \simeq$ $D_6:F_{11}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{66}$	$G/\operatorname{soc} \simeq$ $C_2\times C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
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

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 1320: $D_6:F_{11}$

Order 660: $C_{33}:C_{20}$, $C_{66}:C_{10}$, $C_6\times F_{11}$

Order 440: $D_{22}:C_{10}$

Order 330: $C_{11}:C_{30}$, $C_{33}:C_{10}$, $C_3\times F_{11}$

Order 264: $C_{33}:D_4$

Order 220: $C_{22}:C_{10}$, $C_2\times F_{11}$, $C_{11}:C_{20}$

Order 165: $C_{11}:C_{15}$

Order 132: $S_3\times C_{22}$, $C_3\times D_{22}$, $C_{33}:C_4$

Order 120: $C_{15}:D_4$

Order 110: $C_{11}:C_{10}$ x 2, $F_{11}$

Order 88: $C_{11}:D_4$

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

Order 60: $C_2\times C_{30}$, $S_3\times C_{10}$, $C_3:C_{20}$

Order 55: $C_{11}:C_5$

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

Order 40: $C_5\times D_4$

Order 33: $C_{33}$

Order 30: $C_{30}$ x 2, $C_5\times S_3$

Order 24: $C_3:D_4$

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

Order 20: $C_2\times C_{10}$ x 2, $C_{20}$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 3

Order 8: $D_4$

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

Order 5: $C_5$

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 1320: $D_6:F_{11}$

Order 660: $C_{33}:C_{20}$, $C_{66}:C_{10}$, $C_6\times F_{11}$

Order 440: $D_{22}:C_{10}$

Order 330: $C_{11}:C_{30}$, $C_{33}:C_{10}$, $C_3\times F_{11}$

Order 264: $C_{33}:D_4$

Order 220: $C_{22}:C_{10}$, $C_2\times F_{11}$, $C_{11}:C_{20}$

Order 165: $C_{11}:C_{15}$

Order 132: $S_3\times C_{22}$, $C_3\times D_{22}$, $C_{33}:C_4$

Order 120: $C_{15}:D_4$

Order 110: $C_{11}:C_{10}$ x 2, $F_{11}$

Order 88: $C_{11}:D_4$

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

Order 60: $C_2\times C_{30}$, $S_3\times C_{10}$, $C_3:C_{20}$

Order 55: $C_{11}:C_5$

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

Order 40: $C_5\times D_4$

Order 33: $C_{33}$

Order 30: $C_{30}$ x 2, $C_5\times S_3$

Order 24: $C_3:D_4$

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

Order 20: $C_2\times C_{10}$ x 2, $C_{20}$

Order 15: $C_{15}$

Order 12: $C_2\times C_6$, $D_6$, $C_3:C_4$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 3

Order 8: $D_4$

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

Order 5: $C_5$

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 1320: $D_6:F_{11}$ ($C_1$)

Order 660: $C_{33}:C_{20}$ ($C_2$), $C_{66}:C_{10}$ ($C_2$), $C_6\times F_{11}$ ($C_2$)

Order 330: $C_{11}:C_{30}$ ($C_2^2$)

Order 264: $C_{33}:D_4$ ($C_5$)

Order 220: $C_2\times F_{11}$ ($S_3$)

Order 165: $C_{11}:C_{15}$ ($D_4$)

Order 132: $S_3\times C_{22}$ ($C_{10}$), $C_3\times D_{22}$ ($C_{10}$), $C_{33}:C_4$ ($C_{10}$)

Order 110: $C_{11}:C_{10}$ ($D_6$)

Order 66: $C_{66}$ ($C_2\times C_{10}$)

Order 55: $C_{11}:C_5$ ($C_3:D_4$)

Order 44: $D_{22}$ ($C_5\times S_3$)

Order 33: $C_{33}$ ($C_5\times D_4$)

Order 22: $C_{22}$ ($S_3\times C_{10}$)

Order 12: $D_6$ ($F_{11}$)

Order 11: $C_{11}$ ($C_{15}:D_4$)

Order 6: $C_6$ ($C_2\times F_{11}$)

Order 3: $C_3$ ($D_{22}:C_{10}$)

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

Order 1: $C_1$ ($D_6:F_{11}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1320: $D_6:F_{11}$ ($C_1$)

Order 660: $C_{33}:C_{20}$ ($C_2$), $C_{66}:C_{10}$ ($C_2$), $C_6\times F_{11}$ ($C_2$)

Order 330: $C_{11}:C_{30}$ ($C_2^2$)

Order 264: $C_{33}:D_4$ ($C_5$)

Order 220: $C_2\times F_{11}$ ($S_3$)

Order 165: $C_{11}:C_{15}$ ($D_4$)

Order 132: $S_3\times C_{22}$ ($C_{10}$), $C_3\times D_{22}$ ($C_{10}$), $C_{33}:C_4$ ($C_{10}$)

Order 110: $C_{11}:C_{10}$ ($D_6$)

Order 66: $C_{66}$ ($C_2\times C_{10}$)

Order 55: $C_{11}:C_5$ ($C_3:D_4$)

Order 44: $D_{22}$ ($C_5\times S_3$)

Order 33: $C_{33}$ ($C_5\times D_4$)

Order 22: $C_{22}$ ($S_3\times C_{10}$)

Order 12: $D_6$ ($F_{11}$)

Order 11: $C_{11}$ ($C_{15}:D_4$)

Order 6: $C_6$ ($C_2\times F_{11}$)

Order 3: $C_3$ ($D_{22}:C_{10}$)

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

Order 1: $C_1$ ($D_6:F_{11}$)

Series

Derived series

$D_6:F_{11}$

$\rhd$

$C_{66}$

$\rhd$

$C_1$

Chief series

$D_6:F_{11}$

$\rhd$

$C_{66}:C_{10}$

$\rhd$

$C_{11}:C_{30}$

$\rhd$

$C_{11}:C_{10}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$D_6:F_{11}$

$\rhd$

$C_{66}$

$\rhd$

$C_{33}$

Upper central series

$C_1$

$\lhd$

$C_2$

Character theory

Complex character table

See the $51 \times 51$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $21 \times 21$ rational character table.