Abstract group 19360.h: $C_{11}^2:(C_{10}\times \SD_{16})$

• Label: 19360.h
• Order: \( 2^{5} \cdot 5 \cdot 11^{2} \)
• Exponent: \( 2^{3} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 5 \cdot 11^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $30$
• Trans deg.: $88$
• Rank: $3$

Group information

Description:	$C_{11}^2:(C_{10}\times \SD_{16})$
Order:	\(19360\)\(\medspace = 2^{5} \cdot 5 \cdot 11^{2} \)
Exponent:	\(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^2.C_2^3.C_5.C_2^5$, of order \(154880\)\(\medspace = 2^{8} \cdot 5 \cdot 11^{2} \)
Composition factors:	$C_2$ x 5, $C_5$, $C_{11}$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	4	5	8	10	11	20	22	40	44
Elements	1	331	332	484	968	5324	120	5808	1000	3872	1120	19360
Conjugacy classes	1	5	4	4	4	20	4	16	6	16	8	88
Divisions	1	5	4	1	2	5	4	4	6	2	6	40
Autjugacy classes	1	3	3	4	1	12	4	12	5	4	5	54

Dimension	1	2	4	8	16	20	40	80
Irr. complex chars.	40	30	0	0	0	8	10	0	88
Irr. rational chars.	8	2	10	2	2	8	6	2	40

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$88$
Rank:	$3$
Inequivalent generating triples:	$15249024$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	40	40	40
Arbitrary	22	24	24

Constructions

Presentation:	${\langle a, b, c, d \mid a^{10}=c^{11}=d^{44}=[c,d]=1, b^{4}=d^{22}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,2,8,6,11,5,10,13,18,20,4)(3,7,21,14,15,17,9,19,12,16,22)(23,24)(25,27) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 2 & 7 & 6 & 1 \\ 3 & 7 & 2 & 10 \\ 8 & 1 & 0 & 7 \\ 6 & 9 & 4 & 2 \end{array}\right), \left(\begin{array}{rrrr} 5 & 3 & 8 & 0 \\ 0 & 0 & 1 & 6 \\ 5 & 1 & 4 & 8 \\ 8 & 7 & 5 & 2 \end{array}\right), \left(\begin{array}{rrrr} 9 & 9 & 2 & 3 \\ 3 & 8 & 2 & 2 \\ 9 & 0 & 1 & 2 \\ 1 & 9 & 8 & 0 \end{array}\right), \left(\begin{array}{rrrr} 7 & 10 & 1 & 7 \\ 3 & 8 & 0 & 1 \\ 7 & 8 & 5 & 1 \\ 5 & 7 & 8 & 6 \end{array}\right), \left(\begin{array}{rrrr} 8 & 0 & 2 & 2 \\ 5 & 2 & 5 & 10 \\ 7 & 6 & 4 & 6 \\ 1 & 9 & 9 & 3 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & 5 & 1 \\ 6 & 10 & 6 & 5 \\ 5 & 3 & 10 & 9 \\ 6 & 5 & 5 & 9 \end{array}\right), \left(\begin{array}{rrrr} 4 & 10 & 1 & 7 \\ 4 & 6 & 3 & 1 \\ 5 & 6 & 7 & 1 \\ 8 & 5 & 7 & 9 \end{array}\right), \left(\begin{array}{rrrr} 10 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{44}.D_{22})$ $\,\rtimes\,$ $C_{10}$	$(C_{11}:F_{11})$ $\,\rtimes\,$ $\SD_{16}$ (2)	$(C_{11}^2:C_{40})$ $\,\rtimes\,$ $C_2^2$ (2)	$(C_{11}^2:\SD_{16})$ $\,\rtimes\,$ $C_{10}$ (4)	all 15
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(D_{44}:F_{11})$ . $C_2$	$(C_{22}:F_{11})$ . $D_4$	$(C_{44}:D_{22})$ . $C_{10}$	$(C_{11}^2:C_{20})$ . $D_4$	all 16

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

Homology

Abelianization:	$C_{2}^{2} \times C_{10} \simeq C_{2}^{3} \times C_{5}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 13840 subgroups in 260 conjugacy classes, 47 normal (23 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times D_{11}^2:C_{10}$
Commutator:	$G' \simeq$ $C_{11}^2:C_4$	$G/G' \simeq$ $C_2^2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times D_{11}^2:C_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times C_{44}$	$G/\operatorname{Fit} \simeq$ $C_5\times D_4$
Radical:	$R \simeq$ $C_{11}^2:(C_{10}\times \SD_{16})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}\times C_{22}$	$G/\operatorname{soc} \simeq$ $D_4\times C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times \SD_{16}$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 19360: $C_{11}^2:(C_{10}\times \SD_{16})$

Order 9680: $C_{11}^2:(C_5\times \SD_{16})$ x 4, $C_{11}:(Q_8\times F_{11})$, $D_{44}:F_{11}$, $C_{11}^2:(C_2\times C_{40})$

Order 4840: $C_{11}^2:(C_5\times Q_8)$ x 2, $D_{22}:F_{11}$ x 2, $C_{11}^2:C_{40}$ x 2, $C_{44}:F_{11}$, $D_{22}:F_{11}$, $C_{44}:F_{11}$, $C_{44}.F_{11}$, $C_{11}:(C_4\times F_{11})$

Order 3872: $(C_{11}\times C_{44}).D_4$

Order 2420: $C_{22}:F_{11}$ x 2, $D_{11}:F_{11}$ x 2, $C_{11}^2:C_{20}$ x 2, $C_{11}^2:C_{20}$, $C_{22}:F_{11}$, $C_{11}^2:C_{20}$

Order 1936: $C_{11}^2:\SD_{16}$ x 4, $C_{44}:D_{22}$, $C_{44}.D_{22}$, $C_{11}^2:(C_2\times C_8)$

Order 1210: $C_{11}:F_{11}$ x 2, $C_{11}:F_{11}$ x 2, $C_{11}^2:C_{10}$

Order 968: $C_{11}^2:Q_8$ x 2, $D_{22}:D_{11}$ x 2, $C_{11}^2:C_8$ x 2, $D_{11}\times D_{22}$, $C_{44}:D_{11}$, $C_{11}\times D_{44}$, $C_{11}^2:Q_8$, $C_{22}.D_{22}$

Order 880: $Q_8\times F_{11}$, $D_4\times F_{11}$

Order 605: $C_{11}^2:C_5$

Order 484: $D_{11}^2$ x 2, $C_{11}:C_{44}$ x 2, $C_{11}\times D_{22}$ x 2, $C_{11}\times C_{44}$, $C_{11}^2:C_4$, $C_{11}:D_{22}$

Order 440: $C_4\times F_{11}$ x 6, $C_{44}.C_{10}$ x 3, $C_2^2\times F_{11}$ x 2, $D_{22}:C_{10}$ x 2, $C_{44}:C_{10}$, $C_{44}.C_{10}$, $C_{44}:C_{10}$

Order 242: $C_{11}:D_{11}$ x 2, $C_{11}\times D_{11}$ x 2, $C_{11}\times C_{22}$

Order 220: $C_2\times F_{11}$ x 10, $C_{11}:C_{20}$ x 6, $C_{11}:C_{20}$ x 6, $C_{22}:C_{10}$ x 2

Order 176: $Q_8\times D_{11}$, $D_4\times D_{11}$

Order 160: $C_{10}\times \SD_{16}$

Order 121: $C_{11}^2$

Order 110: $F_{11}$ x 10, $C_{11}:C_{10}$ x 6

Order 88: $C_4\times D_{11}$ x 6, $C_{11}:Q_8$ x 3, $C_{11}:D_4$ x 2, $C_2\times D_{22}$ x 2, $D_4\times C_{11}$, $D_{44}$, $Q_8\times C_{11}$

Order 80: $C_5\times \SD_{16}$ x 4, $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_2\times C_{40}$

Order 55: $C_{11}:C_5$ x 4

Order 44: $D_{22}$ x 10, $C_{44}$ x 6, $C_{11}:C_4$ x 6, $C_2\times C_{22}$ x 2

Order 40: $C_5\times Q_8$ x 3, $C_5\times D_4$ x 3, $C_2\times C_{20}$ x 2, $C_{40}$ x 2, $C_2^2\times C_{10}$

Order 32: $C_2\times \SD_{16}$

Order 22: $D_{11}$ x 10, $C_{22}$ x 6

Order 20: $C_2\times C_{10}$ x 5, $C_{20}$ x 4

Order 16: $\SD_{16}$ x 4, $C_2\times C_8$, $C_2\times Q_8$, $C_2\times D_4$

Order 11: $C_{11}$ x 4

Order 10: $C_{10}$ x 5

Order 8: $Q_8$ x 3, $D_4$ x 3, $C_2\times C_4$ x 2, $C_8$ x 2, $C_2^3$

Order 5: $C_5$

Order 4: $C_2^2$ x 5, $C_4$ x 4

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 19360: $C_{11}^2:(C_{10}\times \SD_{16})$

Order 9680: $C_{11}:(Q_8\times F_{11})$, $D_{44}:F_{11}$, $C_{11}^2:(C_5\times \SD_{16})$, $C_{11}^2:(C_2\times C_{40})$

Order 4840: $C_{44}:F_{11}$, $D_{22}:F_{11}$, $C_{44}:F_{11}$, $C_{44}.F_{11}$, $C_{11}^2:(C_5\times Q_8)$, $C_{11}:(C_4\times F_{11})$, $D_{22}:F_{11}$, $C_{11}^2:C_{40}$

Order 3872: $(C_{11}\times C_{44}).D_4$

Order 2420: $C_{22}:F_{11}$, $D_{11}:F_{11}$, $C_{11}^2:C_{20}$, $C_{22}:F_{11}$, $C_{11}^2:C_{20}$, $C_{11}^2:C_{20}$

Order 1936: $C_{44}:D_{22}$, $C_{44}.D_{22}$, $C_{11}^2:(C_2\times C_8)$, $C_{11}^2:\SD_{16}$

Order 1210: $C_{11}:F_{11}$, $C_{11}^2:C_{10}$, $C_{11}:F_{11}$

Order 968: $D_{11}\times D_{22}$, $C_{44}:D_{11}$, $C_{11}\times D_{44}$, $C_{11}^2:Q_8$, $C_{11}^2:Q_8$, $D_{22}:D_{11}$, $C_{22}.D_{22}$, $C_{11}^2:C_8$

Order 880: $Q_8\times F_{11}$, $D_4\times F_{11}$

Order 605: $C_{11}^2:C_5$

Order 484: $D_{11}^2$, $C_{11}\times C_{44}$, $C_{11}^2:C_4$, $C_{11}:C_{44}$, $C_{11}:D_{22}$, $C_{11}\times D_{22}$

Order 440: $C_4\times F_{11}$ x 5, $C_{44}.C_{10}$ x 2, $C_{44}:C_{10}$, $C_2^2\times F_{11}$, $C_{44}.C_{10}$, $C_{44}:C_{10}$, $D_{22}:C_{10}$

Order 242: $C_{11}\times C_{22}$, $C_{11}:D_{11}$, $C_{11}\times D_{11}$

Order 220: $C_2\times F_{11}$ x 6, $C_{11}:C_{20}$ x 5, $C_{11}:C_{20}$ x 5, $C_{22}:C_{10}$

Order 176: $Q_8\times D_{11}$, $D_4\times D_{11}$

Order 160: $C_{10}\times \SD_{16}$

Order 121: $C_{11}^2$

Order 110: $C_{11}:C_{10}$ x 5, $F_{11}$ x 5

Order 88: $C_4\times D_{11}$ x 5, $C_{11}:Q_8$ x 2, $D_4\times C_{11}$, $C_{11}:D_4$, $D_{44}$, $C_2\times D_{22}$, $Q_8\times C_{11}$

Order 80: $Q_8\times C_{10}$, $D_4\times C_{10}$, $C_5\times \SD_{16}$, $C_2\times C_{40}$

Order 55: $C_{11}:C_5$ x 4

Order 44: $D_{22}$ x 6, $C_{44}$ x 5, $C_{11}:C_4$ x 5, $C_2\times C_{22}$

Order 40: $C_2\times C_{20}$ x 2, $C_5\times Q_8$ x 2, $C_5\times D_4$ x 2, $C_{40}$, $C_2^2\times C_{10}$

Order 32: $C_2\times \SD_{16}$

Order 22: $C_{22}$ x 5, $D_{11}$ x 5

Order 20: $C_2\times C_{10}$ x 3, $C_{20}$ x 3

Order 16: $\SD_{16}$, $C_2\times C_8$, $C_2\times Q_8$, $C_2\times D_4$

Order 11: $C_{11}$ x 4

Order 10: $C_{10}$ x 3

Order 8: $Q_8$ x 2, $D_4$ x 2, $C_2\times C_4$ x 2, $C_2^3$, $C_8$

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 19360: $C_{11}^2:(C_{10}\times \SD_{16})$ ($C_1$)

Order 9680: $C_{11}^2:(C_5\times \SD_{16})$ ($C_2$) x 4, $C_{11}:(Q_8\times F_{11})$ ($C_2$), $D_{44}:F_{11}$ ($C_2$), $C_{11}^2:(C_2\times C_{40})$ ($C_2$)

Order 4840: $C_{11}^2:(C_5\times Q_8)$ ($C_2^2$) x 2, $D_{22}:F_{11}$ ($C_2^2$) x 2, $C_{11}^2:C_{40}$ ($C_2^2$) x 2, $C_{44}:F_{11}$ ($C_2^2$)

Order 3872: $(C_{11}\times C_{44}).D_4$ ($C_5$)

Order 2420: $C_{11}^2:C_{20}$ ($D_4$), $C_{22}:F_{11}$ ($D_4$), $C_{11}^2:C_{20}$ ($C_2^3$)

Order 1936: $C_{11}^2:\SD_{16}$ ($C_{10}$) x 4, $C_{44}:D_{22}$ ($C_{10}$), $C_{44}.D_{22}$ ($C_{10}$), $C_{11}^2:(C_2\times C_8)$ ($C_{10}$)

Order 1210: $C_{11}:F_{11}$ ($\SD_{16}$) x 2, $C_{11}^2:C_{10}$ ($C_2\times D_4$)

Order 968: $C_{11}^2:Q_8$ ($C_2\times C_{10}$) x 2, $D_{22}:D_{11}$ ($C_2\times C_{10}$) x 2, $C_{11}^2:C_8$ ($C_2\times C_{10}$) x 2, $C_{44}:D_{11}$ ($C_2\times C_{10}$)

Order 605: $C_{11}^2:C_5$ ($C_2\times \SD_{16}$)

Order 484: $C_{11}\times C_{44}$ ($C_5\times D_4$), $C_{11}^2:C_4$ ($C_2^2\times C_{10}$), $C_{11}:D_{22}$ ($C_5\times D_4$)

Order 242: $C_{11}:D_{11}$ ($C_5\times \SD_{16}$) x 2, $C_{11}\times C_{22}$ ($D_4\times C_{10}$)

Order 121: $C_{11}^2$ ($C_{10}\times \SD_{16}$)

Order 4: $C_4$ ($D_{11}^2:C_{10}$)

Order 2: $C_2$ ($C_2\times D_{11}^2:C_{10}$)

Order 1: $C_1$ ($C_{11}^2:(C_{10}\times \SD_{16})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 19360: $C_{11}^2:(C_{10}\times \SD_{16})$ ($C_1$)

Order 9680: $C_{11}:(Q_8\times F_{11})$ ($C_2$), $D_{44}:F_{11}$ ($C_2$), $C_{11}^2:(C_5\times \SD_{16})$ ($C_2$), $C_{11}^2:(C_2\times C_{40})$ ($C_2$)

Order 4840: $C_{44}:F_{11}$ ($C_2^2$), $C_{11}^2:(C_5\times Q_8)$ ($C_2^2$), $D_{22}:F_{11}$ ($C_2^2$), $C_{11}^2:C_{40}$ ($C_2^2$)

Order 3872: $(C_{11}\times C_{44}).D_4$ ($C_5$)

Order 2420: $C_{11}^2:C_{20}$ ($D_4$), $C_{22}:F_{11}$ ($D_4$), $C_{11}^2:C_{20}$ ($C_2^3$)

Order 1936: $C_{44}:D_{22}$ ($C_{10}$), $C_{44}.D_{22}$ ($C_{10}$), $C_{11}^2:(C_2\times C_8)$ ($C_{10}$), $C_{11}^2:\SD_{16}$ ($C_{10}$)

Order 1210: $C_{11}:F_{11}$ ($\SD_{16}$), $C_{11}^2:C_{10}$ ($C_2\times D_4$)

Order 968: $C_{44}:D_{11}$ ($C_2\times C_{10}$), $C_{11}^2:Q_8$ ($C_2\times C_{10}$), $D_{22}:D_{11}$ ($C_2\times C_{10}$), $C_{11}^2:C_8$ ($C_2\times C_{10}$)

Order 605: $C_{11}^2:C_5$ ($C_2\times \SD_{16}$)

Order 484: $C_{11}\times C_{44}$ ($C_5\times D_4$), $C_{11}^2:C_4$ ($C_2^2\times C_{10}$), $C_{11}:D_{22}$ ($C_5\times D_4$)

Order 242: $C_{11}\times C_{22}$ ($D_4\times C_{10}$), $C_{11}:D_{11}$ ($C_5\times \SD_{16}$)

Order 121: $C_{11}^2$ ($C_{10}\times \SD_{16}$)

Order 4: $C_4$ ($D_{11}^2:C_{10}$)

Order 2: $C_2$ ($C_2\times D_{11}^2:C_{10}$)

Order 1: $C_1$ ($C_{11}^2:(C_{10}\times \SD_{16})$)

Series

Derived series

$C_{11}^2:(C_{10}\times \SD_{16})$

$\rhd$

$C_{11}^2:(C_{10}\times \SD_{16})$

$\rhd$

$C_{11}^2:C_4$

$\rhd$

$C_{11}^2:C_4$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{11}^2:(C_{10}\times \SD_{16})$

$\rhd$

$C_{11}^2:(C_{10}\times \SD_{16})$

$\rhd$

$C_{11}^2:(C_5\times \SD_{16})$

$\rhd$

$C_{11}^2:(C_5\times \SD_{16})$

$\rhd$

$D_{22}:F_{11}$

$\rhd$

$D_{22}:F_{11}$

$\rhd$

$C_{11}^2:C_{20}$

$\rhd$

$C_{11}^2:C_{20}$

$\rhd$

$C_{11}^2:C_4$

$\rhd$

$C_{11}^2:C_4$

$\rhd$

$C_{11}\times C_{22}$

$\rhd$

$C_{11}\times C_{22}$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{11}^2:(C_{10}\times \SD_{16})$

$\rhd$

$C_{11}^2:(C_{10}\times \SD_{16})$

$\rhd$

$C_{11}^2:C_4$

$\rhd$

$C_{11}^2:C_4$

$\rhd$

$C_{11}\times C_{22}$

$\rhd$

$C_{11}\times C_{22}$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 2 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 $88 \times 88$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.