Abstract group 79860.n: $\He_{11}:C_3:C_{20}$

• Label: 79860.n
• Order: \( 2^{2} \cdot 3 \cdot 5 \cdot 11^{3} \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 5 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3 \cdot 5 \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $726$
• Trans deg.: $726$
• Rank: $2$

Group information

Description:	$\He_{11}:C_3:C_{20}$
Order:	\(79860\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11^{3} \)
Exponent:	\(660\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Automorphism group:	$\He_{11}:C_{15}:Q_8$, of order \(159720\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 11^{3} \)
Composition factors:	$C_2$ x 2, $C_3$, $C_5$, $C_{11}$ x 3
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	5	6	10	11	15	20	22	30	33	44	66
Elements	1	121	242	726	5324	242	5324	1330	10648	31944	1210	10648	2420	7260	2420	79860
Conjugacy classes	1	1	1	2	4	1	4	4	4	8	2	4	2	4	2	44
Divisions	1	1	1	1	1	1	1	3	1	1	1	1	1	1	1	17
Autjugacy classes	1	1	1	1	4	1	4	3	4	4	2	4	2	2	2	36

Dimension	1	2	4	8	55	60	110	220
Irr. complex chars.	20	10	0	0	8	2	4	0	44
Irr. rational chars.	2	3	2	3	0	2	2	3	17

Minimal presentations

Permutation degree:	$726$
Transitive degree:	$726$
Rank:	$2$
Inequivalent generating pairs:	$19008$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	55	110	110
Arbitrary	55	110	110

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{6}=c^{11}=d^{11}=e^{11}=[b,e]=[c,e]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $726$ $\langle(2,9,60,79,51,300,76,85,11,61)(3,18,119,174,249,83,113,178,20,121)(4,27,179,175,617,562,193,228,29,181) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 3 & 0 & 10 & 10 \\ 3 & 0 & 3 & 10 \\ 3 & 8 & 2 & 0 \\ 1 & 3 & 8 & 10 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 7 & 6 & 0 & 0 \\ 3 & 0 & 6 & 0 \\ 8 & 7 & 2 & 3 \end{array}\right), \left(\begin{array}{rrrr} 6 & 3 & 5 & 0 \\ 10 & 8 & 10 & 9 \\ 3 & 1 & 5 & 2 \\ 9 & 5 & 6 & 5 \end{array}\right), \left(\begin{array}{rrrr} 0 & 3 & 6 & 0 \\ 10 & 7 & 3 & 6 \\ 4 & 3 & 6 & 8 \\ 8 & 4 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 9 & 6 & 5 & 3 \\ 9 & 10 & 6 & 7 \\ 4 & 7 & 6 & 0 \\ 10 & 9 & 1 & 9 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 4 & 0 & 0 \\ 4 & 0 & 4 & 0 \\ 5 & 5 & 3 & 5 \end{array}\right), \left(\begin{array}{rrrr} 6 & 9 & 1 & 4 \\ 4 & 6 & 3 & 1 \\ 8 & 10 & 7 & 2 \\ 2 & 8 & 7 & 7 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(\He_{11}:C_{15})$ $\,\rtimes\,$ $C_4$	$(\He_{11}:C_3)$ $\,\rtimes\,$ $C_{20}$	$\He_{11}$ $\,\rtimes\,$ $(C_3:C_{20})$	$(\He_{11}:C_3:C_4)$ $\,\rtimes\,$ $C_5$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_{11}^2:F_{11})$ . $S_3$	$(\He_{11}:C_{30})$ . $C_2$	$(\He_{11}:C_6)$ . $C_{10}$	$(C_{11}^2:D_{11})$ . $(C_5\times S_3)$	all 5

Elements of the group are displayed as matrices in $\GL_{4}(\F_{11})$.

Homology

Abelianization:	$C_{20} \simeq C_{4} \times C_{5}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 16904 subgroups in 52 conjugacy classes, 12 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\He_{11}:C_3:C_{20}$
Commutator:	$G' \simeq$ $\He_{11}:C_3$	$G/G' \simeq$ $C_{20}$
Frattini:	$\Phi \simeq$ $C_{11}$	$G/\Phi \simeq$ $C_{11}^2:(C_5\times C_3:C_4)$
Fitting:	$\operatorname{Fit} \simeq$ $\He_{11}$	$G/\operatorname{Fit} \simeq$ $C_3:C_{20}$
Radical:	$R \simeq$ $\He_{11}:C_3:C_{20}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}$	$G/\operatorname{soc} \simeq$ $C_{11}^2:(C_5\times C_3:C_4)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $\He_{11}$

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 79860: $\He_{11}:C_3:C_{20}$

Order 39930: $\He_{11}:C_{30}$

Order 26620: $\He_{11}:C_{20}$

Order 19965: $\He_{11}:C_{15}$

Order 15972: $\He_{11}:C_3:C_4$

Order 13310: $C_{11}^2:F_{11}$

Order 7986: $\He_{11}:C_6$

Order 6655: $\He_{11}:C_5$

Order 5324: $\He_{11}:C_4$

Order 3993: $\He_{11}:C_3$

Order 2662: $C_{11}^2:D_{11}$

Order 1331: $\He_{11}$

Order 1210: $C_{11}:F_{11}$ x 2

Order 660: $C_{33}:C_{20}$

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

Order 330: $C_{11}:C_{30}$

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

Order 220: $C_{11}:C_{20}$

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

Order 132: $C_3:C_{44}$

Order 121: $C_{11}^2$ x 2

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

Order 66: $C_{66}$

Order 60: $C_3:C_{20}$

Order 55: $C_{11}:C_5$ x 3

Order 44: $C_{44}$

Order 33: $C_{33}$

Order 30: $C_{30}$

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

Order 20: $C_{20}$

Order 15: $C_{15}$

Order 12: $C_3:C_4$

Order 11: $C_{11}$ x 3

Order 10: $C_{10}$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 79860: $\He_{11}:C_3:C_{20}$

Order 39930: $\He_{11}:C_{30}$

Order 26620: $\He_{11}:C_{20}$

Order 19965: $\He_{11}:C_{15}$

Order 15972: $\He_{11}:C_3:C_4$

Order 13310: $C_{11}^2:F_{11}$

Order 7986: $\He_{11}:C_6$

Order 6655: $\He_{11}:C_5$

Order 5324: $\He_{11}:C_4$

Order 3993: $\He_{11}:C_3$

Order 2662: $C_{11}^2:D_{11}$

Order 1331: $\He_{11}$

Order 1210: $C_{11}:F_{11}$

Order 660: $C_{33}:C_{20}$

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

Order 330: $C_{11}:C_{30}$

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

Order 220: $C_{11}:C_{20}$

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

Order 132: $C_3:C_{44}$

Order 121: $C_{11}^2$

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

Order 66: $C_{66}$

Order 60: $C_3:C_{20}$

Order 55: $C_{11}:C_5$ x 2

Order 44: $C_{44}$

Order 33: $C_{33}$

Order 30: $C_{30}$

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

Order 20: $C_{20}$

Order 15: $C_{15}$

Order 12: $C_3:C_4$

Order 11: $C_{11}$ x 2

Order 10: $C_{10}$

Order 6: $C_6$

Order 5: $C_5$

Order 4: $C_4$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 79860: $\He_{11}:C_3:C_{20}$ ($C_1$)

Order 39930: $\He_{11}:C_{30}$ ($C_2$)

Order 19965: $\He_{11}:C_{15}$ ($C_4$)

Order 15972: $\He_{11}:C_3:C_4$ ($C_5$)

Order 13310: $C_{11}^2:F_{11}$ ($S_3$)

Order 7986: $\He_{11}:C_6$ ($C_{10}$)

Order 6655: $\He_{11}:C_5$ ($C_3:C_4$)

Order 3993: $\He_{11}:C_3$ ($C_{20}$)

Order 2662: $C_{11}^2:D_{11}$ ($C_5\times S_3$)

Order 1331: $\He_{11}$ ($C_3:C_{20}$)

Order 11: $C_{11}$ ($C_{11}^2:(C_5\times C_3:C_4)$)

Order 1: $C_1$ ($\He_{11}:C_3:C_{20}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 79860: $\He_{11}:C_3:C_{20}$ ($C_1$)

Order 39930: $\He_{11}:C_{30}$ ($C_2$)

Order 19965: $\He_{11}:C_{15}$ ($C_4$)

Order 15972: $\He_{11}:C_3:C_4$ ($C_5$)

Order 13310: $C_{11}^2:F_{11}$ ($S_3$)

Order 7986: $\He_{11}:C_6$ ($C_{10}$)

Order 6655: $\He_{11}:C_5$ ($C_3:C_4$)

Order 3993: $\He_{11}:C_3$ ($C_{20}$)

Order 2662: $C_{11}^2:D_{11}$ ($C_5\times S_3$)

Order 1331: $\He_{11}$ ($C_3:C_{20}$)

Order 11: $C_{11}$ ($C_{11}^2:(C_5\times C_3:C_4)$)

Order 1: $C_1$ ($\He_{11}:C_3:C_{20}$)

Series

Derived series

$\He_{11}:C_3:C_{20}$

$\rhd$

$\He_{11}:C_3$

$\rhd$

$\He_{11}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Chief series

$\He_{11}:C_3:C_{20}$

$\rhd$

$\He_{11}:C_{30}$

$\rhd$

$\He_{11}:C_{15}$

$\rhd$

$\He_{11}:C_3$

$\rhd$

$\He_{11}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$\He_{11}:C_3:C_{20}$

$\rhd$

$\He_{11}:C_3$

Upper central series

$C_1$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $44 \times 44$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $17 \times 17$ rational character table.