Abstract group 98784.a: $C_7^2.\GL(2,7)$

• Label: 98784.a
• Order: \( 2^{5} \cdot 3^{2} \cdot 7^{3} \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{2} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $49$
• Trans deg.: $49$
• Rank: $2$

Group information

Description:	$C_7^2.\GL(2,7)$
Order:	\(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \)
Exponent:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Automorphism group:	$C_7^2.\GL(2,7)$, of order \(98784\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7^{3} \)
Composition factors:	$C_2$ x 2, $C_3$, $C_7$ x 2, $\PSL(2,7)$
Derived length:	$1$

This group is nonabelian and nonsolvable. Whether it is almost simple has not been computed.

Group statistics

Order	1	2	3	4	6	7	8	12	14	16	21	24	42	48
Elements	1	441	3626	2058	25578	2400	4116	4116	4704	8232	9408	8232	9408	16464	98784
Conjugacy classes	1	2	5	1	13	3	2	2	2	4	4	4	4	8	55
Divisions	1	2	3	1	7	3	1	1	2	1	2	1	2	1	28
Autjugacy classes	1	2	5	1	13	3	2	2	2	4	4	4	4	8	55

Dimension	1	2	6	7	8	12	14	16	24	48	96	288
Irr. complex chars.	6	0	21	6	15	0	0	0	0	6	0	1	55
Irr. rational chars.	2	2	1	2	3	2	2	6	2	3	2	1	28

Minimal presentations

Permutation degree:	$49$
Transitive degree:	$49$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	48	48	48
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\AGL(2,7)$, $\AGammaL(2,7)$
Permutation group:	Degree $49$ $\langle(2,4,5,12,10,20)(3,7,15,16,28,29)(9,18,32,44,46,33)(11,22,37,38,23,40)(13,24,25,39,42,47) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_7^2$ . $\GL(2,7)$	$(C_7:F_7)$ . $\PGL(2,7)$	$(C_7^2.\SL(2,7))$ . $C_6$	$(C_7:D_7.\SO(3,7))$ . $C_3$	all 7
Aut. group:	$\Aut(C_7:D_7)$	$\Aut(C_7^2:C_3)$	$\Aut(C_7:F_7)$	$\Aut(C_7^2:C_6)$	all 10

Elements of the group are displayed as matrices in $\AGL(2,7)$.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 94032 subgroups in 236 conjugacy classes, 9 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_7^2.\GL(2,7)$
Commutator:	$G' \simeq$ $C_7^2.\SL(2,7)$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_7^2.\GL(2,7)$
Fitting:	$\operatorname{Fit} \simeq$ $C_7^2$	$G/\operatorname{Fit} \simeq$ $\GL(2,7)$
Radical:	$R \simeq$ $C_7:F_7$	$G/R \simeq$ $\PGL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_7^2$	$G/\operatorname{soc} \simeq$ $\GL(2,7)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{32}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $\He_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

Order 98784: $C_7^2.\GL(2,7)$

Order 49392: $C_7^2:C_3.\SL(2,7)$

Order 32928: $C_7:D_7.\SO(3,7)$

Order 16464: $C_7^2.\SL(2,7)$

Order 12348: $\He_7:C_6^2$

Order 7056: $C_7:D_7.A_4.C_6$

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

Order 4704: $F_{49}:C_2$

Order 4116: $C_7^2:(C_3\times D_{14})$ x 2, $C_7^2:(C_2\times F_7)$, $C_7^2:(C_2\times F_7)$

Order 3528: $C_7^2:(C_3\times \SL(2,3))$, $F_7\wr C_2$

Order 3087: $\He_7:C_3^2$

Order 2352: $C_7^2:(C_3\times Q_{16})$, $C_7^2:(C_2.S_4)$, $C_7^2:(C_3\times D_8)$, $F_{49}$

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

Order 2016: $\GL(2,7)$

Order 1764: $F_7^2$ x 2, $C_7^2:(C_6\times S_3)$, $C_7^2:C_3:C_{12}$

Order 1568: $C_7^2:\SD_{32}$

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

Order 1176: $C_7^2:(C_3\times Q_8)$, $D_7^2:C_6$, $D_7^2:S_3$, $C_7^2:\SL(2,3)$, $C_7^2:\SL(2,3)$, $C_7^2:C_{24}$

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

Order 1008: $C_3\times \SL(2,7)$

Order 882: $C_7:C_3\times F_7$ x 3, $C_7:(C_3\times F_7)$ x 2, $C_7^2:(C_3\times S_3)$

Order 784: $C_7^2:Q_{16}$, $C_7^2:D_8$, $C_7^2:C_{16}$

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

Order 672: $\SL(2,7):C_2$

Order 588: $D_7\times F_7$ x 3, $D_7:F_7$ x 2, $D_7:F_7$ x 2, $C_7^2:D_6$, $C_7^2:C_{12}$, $C_7^2:C_3:C_4$

Order 441: $C_7^2:C_3^2$ x 2

Order 392: $C_7^2:Q_8$, $D_7\wr C_2$, $C_7^2:C_8$

Order 343: $\He_7$

Order 336: $\SL(2,7)$

Order 294: $C_7:F_7$ x 4, $C_7^2:C_6$ x 3, $C_7\times F_7$ x 3, $C_7:F_7$ x 3, $C_7:F_7$ x 3, $C_7:F_7$ x 2, $C_7:F_7$ x 2, $C_7^2:S_3$

Order 252: $C_6\times F_7$ x 2

Order 196: $D_7^2$ x 2, $C_7^2:C_4$

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

Order 144: $C_6.S_4$

Order 126: $C_3\times F_7$ x 4, $C_{21}:C_6$ x 2

Order 98: $C_7\times D_7$ x 4, $C_7:D_7$ x 2

Order 96: $C_3\times \SD_{32}$

Order 84: $C_2\times F_7$ x 6, $C_3\times D_{14}$ x 2

Order 72: $C_6\wr C_2$, $C_3\times \SL(2,3)$

Order 63: $C_{21}:C_3$ x 2

Order 49: $C_7^2$ x 3

Order 48: $C_2.S_4$, $C_3\times Q_{16}$, $C_3\times D_8$, $C_{48}$

Order 42: $F_7$ x 13, $C_7:C_6$ x 6, $C_3\times D_7$ x 4, $C_{42}$ x 2

Order 36: $C_3:C_{12}$, $C_6^2$, $C_6\times S_3$

Order 32: $\SD_{32}$

Order 28: $D_{14}$ x 2

Order 24: $\SL(2,3)$ x 2, $C_3:D_4$, $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$

Order 21: $C_7:C_3$ x 7, $C_{21}$ x 2

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

Order 16: $Q_{16}$, $D_8$, $C_{16}$

Order 14: $D_7$ x 5, $C_{14}$ x 2

Order 12: $C_2\times C_6$ x 3, $D_6$, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $Q_8$, $D_4$, $C_8$

Order 7: $C_7$ x 3

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 98784: $C_7^2.\GL(2,7)$

Order 49392: $C_7^2:C_3.\SL(2,7)$

Order 32928: $C_7:D_7.\SO(3,7)$

Order 16464: $C_7^2.\SL(2,7)$

Order 12348: $\He_7:C_6^2$

Order 7056: $C_7:D_7.A_4.C_6$

Order 6174: $C_7^2:(C_3\times F_7)$

Order 4704: $F_{49}:C_2$

Order 3087: $\He_7:C_3^2$

Order 2016: $\GL(2,7)$

Order 1568: $C_7^2:\SD_{32}$

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

Order 1008: $C_3\times \SL(2,7)$

Order 672: $\SL(2,7):C_2$

Order 343: $\He_7$

Order 336: $\SL(2,7)$

Order 144: $C_6.S_4$

Order 96: $C_3\times \SD_{32}$

Order 32: $\SD_{32}$

Order 9: $C_3^2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 98784: $C_7^2.\GL(2,7)$ ($C_1$)

Order 49392: $C_7^2:C_3.\SL(2,7)$ ($C_2$)

Order 32928: $C_7:D_7.\SO(3,7)$ ($C_3$)

Order 16464: $C_7^2.\SL(2,7)$ ($C_6$)

Order 294: $C_7:F_7$ ($\PGL(2,7)$)

Order 147: $C_7^2:C_3$ ($\SL(2,7):C_2$)

Order 98: $C_7:D_7$ ($C_3\times \PGL(2,7)$)

Order 49: $C_7^2$ ($\GL(2,7)$)

Order 1: $C_1$ ($C_7^2.\GL(2,7)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 98784: $C_7^2.\GL(2,7)$ ($C_1$)

Order 49392: $C_7^2:C_3.\SL(2,7)$ ($C_2$)

Order 32928: $C_7:D_7.\SO(3,7)$ ($C_3$)

Order 16464: $C_7^2.\SL(2,7)$ ($C_6$)

Order 294: $C_7:F_7$ ($\PGL(2,7)$)

Order 1: $C_1$ ($C_7^2.\GL(2,7)$)

Series

Derived series

$C_7^2.\GL(2,7)$

$\rhd$

$C_7^2.\SL(2,7)$

Chief series

$C_7^2.\GL(2,7)$

$\rhd$

$C_7^2:C_3.\SL(2,7)$

$\rhd$

$C_7:F_7$

$\rhd$

$C_7^2:C_3$

$\rhd$

$C_7^2$

$\rhd$

$C_1$

Lower central series

$C_7^2.\GL(2,7)$

$\rhd$

$C_7^2.\SL(2,7)$

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 2 larger groups in the database.

Character theory

Complex character table

See the $55 \times 55$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $28 \times 28$ rational character table.