Abstract group 37044.e: $\He_7:(C_3^2\times D_6)$

• Label: 37044.e
• Order: \( 2^{2} \cdot 3^{3} \cdot 7^{3} \)
• Exponent: \( 2 \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{3} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $101$
• Trans deg.: $294$
• Rank: $2$

Group information

Description:	$\He_7:(C_3^2\times D_6)$
Order:	\(37044\)\(\medspace = 2^{2} \cdot 3^{3} \cdot 7^{3} \)
Exponent:	\(42\)\(\medspace = 2 \cdot 3 \cdot 7 \)
Automorphism group:	$\He_7.(C_6\times S_3^2).C_2$, of order \(148176\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 7^{3} \)
Composition factors:	$C_2$ x 2, $C_3$ x 3, $C_7$ x 3
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	6	7	14	21	42
Elements	1	343	2942	19502	342	2058	5976	5880	37044
Conjugacy classes	1	3	17	33	4	3	20	12	93
Divisions	1	3	9	17	4	3	9	5	51
Autjugacy classes	1	2	7	10	3	2	7	4	36

Dimension	1	2	4	12	18	24	36	42	84	168
Irr. complex chars.	36	18	0	9	12	0	0	18	0	0	93
Irr. rational chars.	4	18	8	1	4	4	4	2	4	2	51

Minimal presentations

Permutation degree:	$101$
Transitive degree:	$294$
Rank:	$2$
Inequivalent generating pairs:	$1344$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	42	84	84
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{6}=b^{6}=c^{21}=d^{7}=e^{7}=[d,e]=1, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $101$ $\langle(1,2,7,32,69,55,62,48,29,40,30,87,70,95)(3,14,33,91,19,71,35,45,63,57,65,90,22,79) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 4 & 4 & 0 & 0 \\ 5 & 0 & 4 & 0 \\ 2 & 3 & 6 & 1 \end{array}\right), \left(\begin{array}{rrrr} 5 & 0 & 5 & 6 \\ 6 & 5 & 0 & 1 \\ 6 & 5 & 3 & 1 \\ 5 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 4 & 2 & 3 \\ 5 & 6 & 1 & 2 \\ 2 & 4 & 5 & 1 \\ 6 & 2 & 3 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 1 & 6 & 0 & 0 \\ 3 & 0 & 6 & 0 \\ 3 & 2 & 4 & 4 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 1 & 3 \\ 3 & 6 & 5 & 1 \\ 3 & 5 & 3 & 6 \\ 6 & 3 & 4 & 6 \end{array}\right), \left(\begin{array}{rrrr} 1 & 6 & 4 & 4 \\ 4 & 5 & 3 & 4 \\ 6 & 2 & 4 & 1 \\ 2 & 6 & 3 & 1 \end{array}\right), \left(\begin{array}{rrrr} 4 & 5 & 3 & 2 \\ 1 & 5 & 1 & 3 \\ 3 & 5 & 4 & 2 \\ 6 & 3 & 6 & 5 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Direct product:	$C_3$ $\, \times\, $ $(\He_7:(C_6\times S_3))$
Semidirect product:	$(\He_7:D_6)$ $\,\rtimes\,$ $C_3^2$	$(\He_7:C_3^2)$ $\,\rtimes\,$ $D_6$	$\He_7$ $\,\rtimes\,$ $(C_3^2\times D_6)$	$(\He_7:C_3)$ $\,\rtimes\,$ $C_6^2$	all 22
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{21}$ . $(C_7^2:(C_6\times S_3))$	$C_7$ . $(C_7^2:(C_3^2\times D_6))$			more information

Elements of the group are displayed as matrices in $\GL_{4}(\F_{7})$.

Homology

Abelianization:	$C_{6}^{2} \simeq C_{2}^{2} \times C_{3}^{2}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 53984 subgroups in 460 conjugacy classes, 46 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $\He_7:(C_6\times S_3)$
Commutator:	$G' \simeq$ $\He_7:C_3$	$G/G' \simeq$ $C_6^2$
Frattini:	$\Phi \simeq$ $C_7$	$G/\Phi \simeq$ $C_7^2:(C_3^2\times D_6)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times \He_7$	$G/\operatorname{Fit} \simeq$ $C_6\times S_3$
Radical:	$R \simeq$ $\He_7:(C_3^2\times D_6)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{21}$	$G/\operatorname{soc} \simeq$ $C_7^2:(C_6\times S_3)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $\He_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

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Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 37044: $\He_7:(C_3^2\times D_6)$

Order 18522: $\He_7:(S_3\times C_3^2)$ x 2, $C_7^2:(C_3^2\times F_7)$

Order 12348: $\He_7:(C_6\times S_3)$ x 3, $\He_7:C_6^2$, $C_3\times \He_7:D_6$

Order 9261: $\He_7:C_3^3$

Order 6174: $C_7^2:(C_3\times F_7)$ x 6, $\He_7:(C_3\times S_3)$ x 6, $C_3\times C_7^2:F_7$ x 2, $C_3\times \He_7:S_3$ x 2, $C_3\times C_7^2:F_7$, $C_3\times C_7^2:F_7$, $C_3\times C_7^2:F_7$

Order 4116: $C_7^2:(C_2\times F_7)$ x 3, $\He_7:D_6$, $C_{21}.D_7^2$

Order 3087: $\He_7:C_3^2$ x 6, $\He_7:C_3^2$, $\He_7:C_3^2$, $\He_7:C_3^2$

Order 2646: $C_{21}:C_3\times F_7$

Order 2058: $C_7^2:F_7$ x 6, $C_7^2:F_7$ x 3, $C_7^2:F_7$ x 3, $\He_7:S_3$ x 2, $C_7^2:C_{42}$ x 2, $C_7^2:F_7$ x 2, $\He_7:C_6$

Order 1764: $C_7^2:C_6^2$ x 2

Order 1372: $C_7^2:D_{14}$

Order 1323: $C_7^2:C_3^3$

Order 1029: $C_7^2:C_{21}$ x 3, $\He_7:C_3$ x 3, $\He_7:C_3$ x 2, $C_3\times \He_7$

Order 882: $C_7:C_3\times F_7$ x 9, $C_{21}:F_7$ x 5, $C_{21}:F_7$ x 2, $C_{21}:F_7$, $C_{21}:C_3\times D_7$, $C_{21}\times F_7$

Order 756: $C_{21}:C_6^2$

Order 686: $C_7^2:C_{14}$ x 2, $C_7^2:D_7$

Order 588: $D_7:F_7$ x 6, $C_3\times D_7^2$ x 2

Order 441: $C_7^2:C_3^2$ x 9, $C_7^2:C_3^2$ x 3, $C_{21}:C_{21}$ x 2, $C_7^2:C_3^2$

Order 378: $C_3^2:F_7$ x 2, $C_{14}:C_3^3$, $C_3^2\times F_7$

Order 343: $\He_7$

Order 294: $C_7:F_7$ x 15, $C_7:F_7$ x 6, $D_7\times C_{21}$ x 5, $C_7^2:C_6$ x 3, $C_7\times F_7$ x 3, $C_7:F_7$ x 3, $C_{21}:D_7$ x 2

Order 252: $C_{42}:C_6$ x 3, $C_6\times F_7$ x 3, $C_3\times D_{42}$

Order 196: $D_7^2$ x 2

Order 189: $C_7:C_3^3$ x 2

Order 147: $C_7^2:C_3$ x 9, $C_7:C_{21}$ x 6, $C_7\times C_{21}$ x 3, $C_7^2:C_3$ x 3

Order 126: $C_3\times F_7$ x 18, $C_{21}:C_6$ x 10, $C_{21}:C_6$ x 6, $C_3\times D_{21}$ x 2, $C_3\times C_{42}$, $C_3^2\times D_7$

Order 108: $C_3^2\times D_6$

Order 98: $C_7\times D_7$ x 5, $C_7:D_7$ x 2

Order 84: $C_2\times F_7$ x 9, $C_3\times D_{14}$ x 3, $D_{42}$

Order 63: $C_{21}:C_3$ x 22, $C_3\times C_{21}$ x 2

Order 54: $S_3\times C_3^2$ x 2, $C_3^2\times C_6$

Order 49: $C_7^2$ x 3

Order 42: $F_7$ x 27, $C_7:C_6$ x 12, $C_3\times D_7$ x 10, $C_{42}$ x 5, $D_{21}$ x 2

Order 36: $C_6\times S_3$ x 4, $C_6^2$

Order 28: $D_{14}$ x 3

Order 27: $C_3^3$

Order 21: $C_7:C_3$ x 21, $C_{21}$ x 9

Order 18: $C_3\times C_6$ x 11, $C_3\times S_3$ x 8

Order 14: $D_7$ x 7, $C_{14}$ x 3

Order 12: $C_2\times C_6$ x 4, $D_6$

Order 9: $C_3^2$ x 9

Order 7: $C_7$ x 4

Order 6: $C_6$ x 17, $S_3$ x 2

Order 4: $C_2^2$

Order 3: $C_3$ x 9

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 37044: $\He_7:(C_3^2\times D_6)$

Order 18522: $C_7^2:(C_3^2\times F_7)$, $\He_7:(S_3\times C_3^2)$

Order 12348: $\He_7:C_6^2$, $C_3\times \He_7:D_6$, $\He_7:(C_6\times S_3)$

Order 9261: $\He_7:C_3^3$

Order 6174: $C_7^2:(C_3\times F_7)$ x 2, $C_3\times C_7^2:F_7$, $C_3\times C_7^2:F_7$, $C_3\times C_7^2:F_7$, $C_3\times C_7^2:F_7$, $\He_7:(C_3\times S_3)$, $C_3\times \He_7:S_3$

Order 4116: $\He_7:D_6$, $C_7^2:(C_2\times F_7)$, $C_{21}.D_7^2$

Order 3087: $\He_7:C_3^2$ x 2, $\He_7:C_3^2$, $\He_7:C_3^2$, $\He_7:C_3^2$

Order 2646: $C_{21}:C_3\times F_7$

Order 2058: $C_7^2:F_7$ x 2, $\He_7:C_6$, $C_7^2:F_7$, $C_7^2:F_7$, $C_7^2:F_7$, $\He_7:S_3$, $C_7^2:C_{42}$

Order 1764: $C_7^2:C_6^2$

Order 1372: $C_7^2:D_{14}$

Order 1323: $C_7^2:C_3^3$

Order 1029: $\He_7:C_3$ x 2, $C_3\times \He_7$, $C_7^2:C_{21}$, $\He_7:C_3$

Order 882: $C_{21}:F_7$ x 3, $C_7:C_3\times F_7$ x 2, $C_{21}:F_7$, $C_{21}:F_7$, $C_{21}:C_3\times D_7$, $C_{21}\times F_7$

Order 756: $C_{21}:C_6^2$

Order 686: $C_7^2:D_7$, $C_7^2:C_{14}$

Order 588: $C_3\times D_7^2$, $D_7:F_7$

Order 441: $C_7^2:C_3^2$ x 2, $C_7^2:C_3^2$ x 2, $C_{21}:C_{21}$ x 2, $C_7^2:C_3^2$

Order 378: $C_{14}:C_3^3$, $C_3^2:F_7$, $C_3^2\times F_7$

Order 343: $\He_7$

Order 294: $D_7\times C_{21}$ x 3, $C_7:F_7$ x 3, $C_7\times F_7$ x 2, $C_7^2:C_6$, $C_{21}:D_7$, $C_7:F_7$, $C_7:F_7$

Order 252: $C_6\times F_7$ x 2, $C_3\times D_{42}$, $C_{42}:C_6$

Order 196: $D_7^2$

Order 189: $C_7:C_3^3$ x 2

Order 147: $C_7:C_{21}$ x 3, $C_7\times C_{21}$ x 2, $C_7^2:C_3$ x 2, $C_7^2:C_3$

Order 126: $C_3\times F_7$ x 8, $C_{21}:C_6$ x 5, $C_{21}:C_6$, $C_3\times C_{42}$, $C_3\times D_{21}$, $C_3^2\times D_7$

Order 108: $C_3^2\times D_6$

Order 98: $C_7\times D_7$ x 3, $C_7:D_7$

Order 84: $C_2\times F_7$ x 2, $C_3\times D_{14}$ x 2, $D_{42}$

Order 63: $C_{21}:C_3$ x 10, $C_3\times C_{21}$ x 2

Order 54: $C_3^2\times C_6$, $S_3\times C_3^2$

Order 49: $C_7^2$ x 2

Order 42: $F_7$ x 7, $C_3\times D_7$ x 5, $C_{42}$ x 4, $C_7:C_6$ x 3, $D_{21}$

Order 36: $C_6\times S_3$ x 2, $C_6^2$

Order 28: $D_{14}$ x 2

Order 27: $C_3^3$

Order 21: $C_7:C_3$ x 7, $C_{21}$ x 6

Order 18: $C_3\times C_6$ x 6, $C_3\times S_3$ x 2

Order 14: $D_7$ x 4, $C_{14}$ x 2

Order 12: $C_2\times C_6$ x 2, $D_6$

Order 9: $C_3^2$ x 5

Order 7: $C_7$ x 3

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

Order 4: $C_2^2$

Order 3: $C_3$ x 5

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 37044: $\He_7:(C_3^2\times D_6)$ ($C_1$)

Order 18522: $\He_7:(S_3\times C_3^2)$ ($C_2$) x 2, $C_7^2:(C_3^2\times F_7)$ ($C_2$)

Order 12348: $\He_7:(C_6\times S_3)$ ($C_3$) x 3, $C_3\times \He_7:D_6$ ($C_3$)

Order 9261: $\He_7:C_3^3$ ($C_2^2$)

Order 6174: $\He_7:(C_3\times S_3)$ ($C_6$) x 6, $C_7^2:(C_3\times F_7)$ ($C_6$) x 3, $C_3\times \He_7:S_3$ ($C_6$) x 2, $C_3\times C_7^2:F_7$ ($C_6$), $C_3\times C_7^2:F_7$ ($S_3$)

Order 4116: $\He_7:D_6$ ($C_3^2$)

Order 3087: $\He_7:C_3^2$ ($C_2\times C_6$) x 3, $\He_7:C_3^2$ ($D_6$), $\He_7:C_3^2$ ($C_2\times C_6$)

Order 2058: $C_7^2:F_7$ ($C_3\times S_3$) x 3, $\He_7:S_3$ ($C_3\times C_6$) x 2, $\He_7:C_6$ ($C_3\times S_3$), $C_7^2:F_7$ ($C_3\times C_6$)

Order 1029: $\He_7:C_3$ ($C_6\times S_3$) x 3, $C_3\times \He_7$ ($C_6\times S_3$), $\He_7:C_3$ ($C_6^2$)

Order 686: $C_7^2:D_7$ ($S_3\times C_3^2$)

Order 343: $\He_7$ ($C_3^2\times D_6$)

Order 21: $C_{21}$ ($C_7^2:(C_6\times S_3)$)

Order 7: $C_7$ ($C_7^2:(C_3^2\times D_6)$)

Order 3: $C_3$ ($\He_7:(C_6\times S_3)$)

Order 1: $C_1$ ($\He_7:(C_3^2\times D_6)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 37044: $\He_7:(C_3^2\times D_6)$ ($C_1$)

Order 18522: $C_7^2:(C_3^2\times F_7)$ ($C_2$), $\He_7:(S_3\times C_3^2)$ ($C_2$)

Order 12348: $C_3\times \He_7:D_6$ ($C_3$), $\He_7:(C_6\times S_3)$ ($C_3$)

Order 9261: $\He_7:C_3^3$ ($C_2^2$)

Order 6174: $C_3\times C_7^2:F_7$ ($C_6$), $C_7^2:(C_3\times F_7)$ ($C_6$), $C_3\times C_7^2:F_7$ ($S_3$), $\He_7:(C_3\times S_3)$ ($C_6$), $C_3\times \He_7:S_3$ ($C_6$)

Order 4116: $\He_7:D_6$ ($C_3^2$)

Order 3087: $\He_7:C_3^2$ ($C_2\times C_6$), $\He_7:C_3^2$ ($D_6$), $\He_7:C_3^2$ ($C_2\times C_6$)

Order 2058: $\He_7:C_6$ ($C_3\times S_3$), $C_7^2:F_7$ ($C_3\times S_3$), $\He_7:S_3$ ($C_3\times C_6$), $C_7^2:F_7$ ($C_3\times C_6$)

Order 1029: $C_3\times \He_7$ ($C_6\times S_3$), $\He_7:C_3$ ($C_6^2$), $\He_7:C_3$ ($C_6\times S_3$)

Order 686: $C_7^2:D_7$ ($S_3\times C_3^2$)

Order 343: $\He_7$ ($C_3^2\times D_6$)

Order 21: $C_{21}$ ($C_7^2:(C_6\times S_3)$)

Order 7: $C_7$ ($C_7^2:(C_3^2\times D_6)$)

Order 3: $C_3$ ($\He_7:(C_6\times S_3)$)

Order 1: $C_1$ ($\He_7:(C_3^2\times D_6)$)

Series

Derived series

$\He_7:(C_3^2\times D_6)$

$\rhd$

$\He_7:(C_3^2\times D_6)$

$\rhd$

$\He_7:C_3$

$\rhd$

$\He_7:C_3$

$\rhd$

$\He_7$

$\rhd$

$\He_7$

$\rhd$

$C_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$\He_7:(C_3^2\times D_6)$

$\rhd$

$\He_7:(C_3^2\times D_6)$

$\rhd$

$\He_7:(S_3\times C_3^2)$

$\rhd$

$\He_7:(S_3\times C_3^2)$

$\rhd$

$\He_7:C_3^3$

$\rhd$

$\He_7:C_3^3$

$\rhd$

$\He_7:C_3^2$

$\rhd$

$\He_7:C_3^2$

$\rhd$

$\He_7:C_3$

$\rhd$

$\He_7:C_3$

$\rhd$

$\He_7$

$\rhd$

$\He_7$

$\rhd$

$C_7$

$\rhd$

$C_7$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$\He_7:(C_3^2\times D_6)$

$\rhd$

$\He_7:(C_3^2\times D_6)$

$\rhd$

$\He_7:C_3$

$\rhd$

$\He_7:C_3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_3$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 1 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 $93 \times 93$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $51 \times 51$ rational character table.