Abstract group 451584.a: $\SL(2,7)^2.C_2^2$

• Label: 451584.a
• Order: \( 2^{10} \cdot 3^{2} \cdot 7^{2} \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $32$
• Trans deg.: $32$
• Rank: $2$

Group information

Description:	$\SL(2,7)^2.C_2^2$
Order:	\(451584\)\(\medspace = 2^{10} \cdot 3^{2} \cdot 7^{2} \)
Exponent:	\(336\)\(\medspace = 2^{4} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^2.\PSL(2,7)^2.D_4$, of order \(903168\)\(\medspace = 2^{11} \cdot 3^{2} \cdot 7^{2} \)
Composition factors:	$C_2$ x 4, $\PSL(2,7)$ x 2
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	28	42	48	56
Elements	1	3811	3248	2604	72464	2400	42672	47040	39456	103488	5376	18816	40320	16128	37632	16128	451584
Conjugacy classes	1	5	2	5	12	3	11	4	9	18	1	4	4	3	8	4	94
Divisions	1	5	2	5	10	3	7	4	9	6	1	2	4	3	1	2	65
Autjugacy classes	1	4	2	4	9	2	10	3	6	10	1	4	3	3	4	4	70

Dimension	1	12	14	16	18	24	32	36	48	49	64	72	84	96	112	128	144	168	192	224	256	288	336	384
Irr. complex chars.	4	7	2	5	4	0	4	12	2	4	8	9	7	17	5	4	0	0	0	0	0	0	0	0	94
Irr. rational chars.	4	1	2	3	4	1	5	4	3	4	8	5	1	3	3	2	2	1	3	1	1	1	1	2	65

Minimal presentations

Permutation degree:	$32$
Transitive degree:	$32$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $32$ $\langle(1,2,7,6,9,26)(3,13,25,16,14,27)(4,10,19,11,20,32)(8,24,21,29,15,28)(12,23) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 5 & 0 & 0 & 0 \\ 5 & 6 & 3 & 0 \\ 6 & 4 & 3 & 0 \\ 2 & 6 & 2 & 4 \end{array}\right), \left(\begin{array}{rrrr} 0 & 2 & 0 & 5 \\ 1 & 1 & 6 & 1 \\ 6 & 4 & 1 & 2 \\ 6 & 3 & 5 & 5 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 4 & 4 & 0 & 0 \\ 2 & 1 & 2 & 0 \\ 3 & 2 & 3 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 5 & 2 & 0 \\ 1 & 6 & 0 & 5 \\ 1 & 0 & 6 & 5 \\ 0 & 6 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 6 & 0 & 0 & 0 \\ 0 & 6 & 0 & 0 \\ 0 & 0 & 6 & 0 \\ 0 & 0 & 0 & 6 \end{array}\right), \left(\begin{array}{rrrr} 3 & 4 & 2 & 3 \\ 3 & 6 & 4 & 4 \\ 3 & 5 & 0 & 4 \\ 4 & 4 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 3 & 1 & 0 \\ 5 & 1 & 2 & 2 \\ 6 & 2 & 5 & 1 \\ 0 & 3 & 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{7})$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$\SL(2,7)^2$ . $C_2^2$	$C_2^2$ . $\POPlus(4,7)$	$C_2$ . $\GOrthPlus(4,7)$	$(\SL(2,7)\wr C_2)$ . $C_2$ (2)	all 5

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 8 normal subgroups (6 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	a subgroup isomorphic to $\SL(2,7)^2$
Frattini:	a subgroup isomorphic to $C_2^2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $Q_{16}^2:C_2^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

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

All subgroups of index up to 32256 (order at least 14) are shown, as well as all normal subgroups of any index.

Order 451584: $\SL(2,7)^2.C_2^2$

Order 225792: $\SL(2,7)\wr C_2$ x 2, $C_2.\SL(2,7).\SO(3,7)$

Order 112896: $\SL(2,7)^2$

Order 28224: $C_{14}:\GL(2,7)$

Order 16128: $(C_2\times \SL(2,7)).S_4$

Order 14112: $C_7:\GL(2,7)$ x 2, $C_{14}.(C_6\times \GL(3,2))$

Order 10752: $\SL(2,7):\SD_{32}$

Order 9408: $\SL(2,7):D_{14}$

Order 8064: $\SL(2,3).\SL(2,7)$, $(C_2\times \GL(2,7)):C_2$

Order 7056: $C_7:C_3\times \SL(2,7)$, $C_{14}^2:(C_6\times S_3)$

Order 5376: $\SL(2,7):C_{16}$, $\SL(2,7):D_8$, $Q_{16}.\SL(2,7)$

Order 4704: $\SL(2,7):D_7$ x 2, $C_{14}\times \SL(2,7)$

Order 4608: $\SL(2,3)^2.D_4$ x 2

Order 4032: $\SL(2,7):D_6$, $C_3:C_4\times \SL(2,7)$, $C_2\times \GL(2,7)$

Order 3528: $C_{14}^2:(C_3\times S_3)$ x 2, $C_7^2:C_6\wr C_2$ x 2, $C_7:F_7:C_{12}$, $C_7^2:(C_6\times D_6)$, $C_2\times C_7^2:C_6^2$

Order 2688: $D_4.\PGL(2,7)$ x 2, $C_8\times \SL(2,7)$, $Q_8\times \SL(2,7)$, $\SL(2,7):D_4$

Order 2352: $C_{14}^2:D_6$, $C_7\times \SL(2,7)$, $C_{14}^2:(C_2\times C_6)$

Order 2304: $\SL(2,3)^2.C_4$ x 2, $Q_8^2.S_3^2$ x 2, $C_2^2.S_4^2$ x 2

Order 2016: $C_6\times \SL(2,7)$, $\SL(2,7):S_3$, $C_{14}.(C_6\times S_4)$, $\GL(2,7)$

Order 1764: $C_7^2:C_6^2$ x 4, $C_7^2:C_3:C_{12}$ x 2, $C_7^2:(C_6\times S_3)$ x 2, $C_7^2:(C_6\times S_3)$ x 2, $C_7^2:C_6^2$

Order 1344: $\SL(2,7):C_2^2$ x 5, $C_2^2.\PGL(2,7)$ x 4, $D_4.\PSL(2,7)$ x 2, $C_4.\PGL(2,7)$ x 2, $\SL(2,7):C_2^2$ x 2, $C_4\times \SL(2,7)$, $C_2\times Q_{16}:F_7$

Order 1176: $D_{14}:F_7$ x 2, $(C_7\times C_{14}).D_6$ x 2, $C_{14}^2:C_6$ x 2, $C_{14}^2:S_3$ x 2, $C_7:(C_4\times F_7)$, $(C_7\times C_{14}).D_6$, $C_{14}^2:C_6$, $C_{14}^2:C_6$, $C_{14}^2:C_6$, $D_{14}:F_7$, $C_2\times C_7^2:D_6$

Order 1152: $\SL(2,3)^2.C_2$ x 2, $\SL(2,3)\wr C_2$ x 2, $\SL(2,3)^2.C_2$ x 2

Order 1024: $Q_{16}^2:C_2^2$

Order 1008: $C_{14}.(C_3\times S_4)$ x 2, $C_7:C_6\times \SL(2,3)$, $C_2\times D_{42}:C_6$, $C_3\times \SL(2,7)$

Order 882: $C_7:(C_3\times F_7)$ x 3, $C_2\times C_7^2:C_3^2$ x 2, $C_7^2:(C_3\times S_3)$ x 2

Order 784: $D_{14}:D_{14}$

Order 768: $Q_8^2.D_6$ x 2, $Q_8^2.D_6$

Order 672: $\SL(2,7):C_2$ x 7, $C_2\times \SL(2,7)$ x 5, $Q_{16}:F_7$ x 4, $\SL(2,7):C_2$ x 2, $C_2.\PGL(2,7)$ x 2, $C_7:C_6\times Q_{16}$, $D_{56}:C_6$, $C_{14}:C_{48}$, $C_2\times C_{14}.S_4$

Order 588: $C_{14}:F_7$ x 6, $C_{14}:F_7$ x 4, $C_{14}:F_7$ x 4, $C_{14}:F_7$ x 2, $C_7^2:D_6$ x 2, $D_7:F_7$ x 2, $C_7^2:D_6$ x 2, $C_7^2:C_{12}$ x 2, $(C_7\times C_{14}).S_3$ x 2, $C_{14}^2:C_3$, $C_{14}^2:C_3$, $C_{14}:C_{42}$

Order 576: $\SL(2,3)^2$ x 2, $C_2^3.\SOPlus(4,2)$, $C_6.(C_4\times S_4)$

Order 512: $Q_{16}\wr C_2$ x 2, $C_8^2.D_4$ x 2, $D_{16}:D_8$, $Q_{16}^2:C_2$, $(C_8\times C_{16}).C_4$

Order 504: $D_{42}:C_6$ x 4, $C_{42}:C_{12}$, $C_7:C_3\times \SL(2,3)$, $D_{42}:C_6$, $C_{14}:C_6^2$

Order 448: $Q_{16}:D_{14}$

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

Order 392: $C_{14}\wr C_2$ x 2, $C_7:D_{28}$ x 2, $C_{14}:D_{14}$, $D_7\times D_{14}$, $C_{14}.D_{14}$

Order 384: $Q_8^2:C_6$ x 2, $\SL(2,3):Q_{16}$ x 2, $Q_8^2:S_3$ x 2, $Q_8^2.S_3$ x 2, $Q_8^2.S_3$ x 2, $Q_{16}\times \SL(2,3)$, $\SL(2,3).Q_{16}$, $C_8\times C_2.S_4$, $(C_3\times Q_{16}):D_4$

Order 336: $C_{56}.C_6$ x 3, $\SL(2,7)$ x 3, $C_{56}:C_6$ x 3, $D_4\times F_7$ x 2, $C_{14}.S_4$ x 2, $C_7:C_{48}$ x 2, $C_{14}:C_{24}$, $C_{14}:\SL(2,3)$, $C_{14}\times \SL(2,3)$, $C_7:C_6\times Q_8$, $C_6:D_{28}$, $D_{28}:C_6$

Order 294: $C_7:F_7$ x 4, $C_7:C_{42}$ x 3, $C_7:F_7$ x 3, $C_7:F_7$ x 3, $C_7^2:S_3$ x 2, $C_7^2:C_6$ x 2, $C_7^2:C_6$ x 2, $C_7:F_7$ x 2

Order 288: $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_2^3:S_3^2$, $C_2\times C_6.S_4$, $C_2^3.S_3^2$, $(C_2\times C_6^2).C_4$, $C_3:C_4\times \SL(2,3)$, $C_6.\GL(2,3)$

Order 256: $Q_8^2.C_2^2$ x 2, $C_{16}.D_8$ x 2, $D_8.D_8$ x 2, $C_8^2.C_4$ x 2, $Q_8^2.C_4$ x 2, $D_{16}:D_4$ x 2, $C_8.Q_{32}$, $Q_{16}:C_{16}$, $C_{16}:D_8$, $Q_{16}:D_8$, $D_8:D_8$, $D_{16}:C_8$, $Q_{16}^2$

Order 252: $C_{42}:C_6$ x 4, $C_6\times F_7$ x 4, $C_{21}:C_{12}$ x 2, $C_{42}:C_6$

Order 224: $Q_{16}:D_7$ x 4, $C_2\times D_{56}$, $C_{14}:C_{16}$, $C_{14}\times Q_{16}$

Order 196: $C_7:D_{14}$ x 4, $D_7^2$ x 2, $C_7:C_{28}$ x 2, $C_7\times D_{14}$ x 2, $C_{14}^2$

Order 192: $\SL(2,3):Q_8$ x 2, $D_4.S_4$ x 2, $Q_8:\SL(2,3)$ x 2, $Q_8\times \SL(2,3)$ x 2, $C_4\times C_2.S_4$, $C_6\times \SD_{32}$, $C_3:C_4\times Q_{16}$, $Q_{16}:D_6$, $C_{24}:D_4$, $D_{24}:C_4$, $C_{24}.D_4$, $C_8\times \SL(2,3)$, $\SL(2,3):C_8$, $C_6.D_{16}$

Order 168: $C_2^2\times F_7$ x 6, $C_{28}:C_6$ x 5, $C_3:D_{28}$ x 4, $D_{14}:C_6$ x 4, $C_4\times F_7$ x 2, $C_{28}.C_6$ x 2, $C_{28}:C_6$ x 2, $C_7:C_{24}$ x 2, $C_2\times D_{42}$, $C_6\times D_{14}$, $C_6:C_{28}$, $C_7:\SL(2,3)$, $C_7\times \SL(2,3)$, $C_{14}:C_{12}$

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

Order 144: $C_6^2:C_2^2$ x 2, $D_6:D_6$ x 2, $D_6.D_6$ x 2, $D_6.D_6$ x 2, $C_3^2:\OD_{16}$ x 2, $C_6.S_4$ x 2, $C_3^2:\SD_{16}$ x 2, $C_3^2:D_8$ x 2, $C_3^2:C_4^2$, $C_6^2:C_2^2$, $C_6\times \SL(2,3)$, $D_6:D_6$, $C_6:D_{12}$, $D_6:D_6$

Order 128: $C_{16}:D_4$ x 2, $C_{16}.D_4$ x 2, $C_8.D_8$ x 2, $D_8:D_4$ x 2, $Q_8\wr C_2$ x 2, $D_4.D_8$ x 2, $D_4.D_8$ x 2, $C_{16}.D_4$ x 2, $C_8.D_8$ x 2, $C_8\wr C_2$ x 2, $Q_8:Q_{16}$ x 2, $D_{16}:C_2^2$ x 2, $C_2.D_4^2$ x 2, $(C_4\times C_8).C_4$ x 2, $C_4^2.D_4$ x 2, $Q_{16}:D_4$, $C_4.D_{16}$, $D_{16}:C_4$, $C_8.D_8$, $C_8:Q_{16}$, $C_8:D_8$, $C_8\times C_{16}$, $C_8.Q_{16}$, $C_8\times Q_{16}$, $Q_8\times Q_{16}$, $D_8:D_4$, $C_{16}.C_8$, $C_8:C_{16}$

Order 126: $C_{21}:C_6$ x 3, $C_{21}:C_6$ x 2, $C_3\times F_7$ x 2

Order 112: $D_{56}$ x 3, $C_7\times Q_{16}$ x 3, $D_4\times D_7$ x 2, $C_7:C_{16}$ x 2, $Q_8\times C_{14}$, $C_2\times D_{28}$, $C_2\times C_{56}$

Order 98: $C_7:D_7$ x 3, $C_7\times C_{14}$ x 2, $C_7\times D_7$ x 2

Order 96: $C_2^2.S_4$ x 5, $\GL(2,3):C_2$ x 4, $\SL(2,3):C_4$ x 2, $C_3\times \SD_{32}$ x 2, $Q_{16}:S_3$ x 2, $D_4.A_4$ x 2, $\GL(2,3):C_2$ x 2, $C_4.S_4$ x 2, $C_4\times \SL(2,3)$, $C_2\times C_{48}$, $C_6.Q_{16}$, $C_{24}:C_4$, $D_4:D_6$, $C_{24}:C_4$, $C_6\times Q_{16}$, $C_6:C_{16}$, $C_6\times D_8$, $C_3:C_4\times Q_8$, $C_{12}:D_4$, $C_6:D_8$, $C_2\times D_{24}$

Order 84: $C_2\times F_7$ x 24, $C_{14}:C_6$ x 6, $C_7:C_{12}$ x 4, $D_{42}$ x 4, $C_3\times D_{14}$ x 4, $C_3:C_{28}$ x 2, $C_7:C_{12}$ x 2, $C_2\times C_{42}$

Order 72: $C_6\wr C_2$ x 6, $C_3:D_{12}$ x 4, $D_6:S_3$ x 4, $C_6\times D_6$ x 2, $C_6^2:C_2$ x 2, $C_6:C_{12}$ x 2, $C_3^2:Q_8$ x 2, $C_6.D_6$ x 2, $C_3^2:C_8$ x 2, $C_2\times C_6^2$, $C_6:D_6$, $S_3\times D_6$, $C_6.D_6$, $C_3\times \SL(2,3)$, $C_6.D_6$

Order 64: $C_2\times \SD_{32}$ x 7, $Q_{32}:C_2$ x 4, $D_{16}:C_2$ x 4, $D_{16}:C_2$ x 3, $C_8.C_8$ x 2, $C_4.D_8$ x 2, $\OD_{32}:C_2$ x 2, $C_4.D_8$ x 2, $C_4^2.C_4$ x 2, $\OD_{32}:C_2$ x 2, $D_4.D_4$ x 2, $D_8:C_2^2$ x 2, $Q_8^2$ x 2, $\OD_{32}:C_2$ x 2, $C_4:Q_{16}$ x 2, $C_4:D_8$ x 2, $Q_8:Q_8$ x 2, $C_8.D_4$ x 2, $C_8.D_4$ x 2, $C_4.Q_{16}$ x 2, $D_4.D_4$ x 2, $D_8:C_4$ x 2, $Q_8:C_8$, $C_{16}.C_4$, $C_4:C_{16}$, $D_8:C_4$, $C_4\times C_{16}$, $C_8^2$, $C_8:Q_8$, $C_8:C_8$, $D_4:D_4$, $C_8\times Q_8$, $C_4\times Q_{16}$

Order 63: $C_{21}:C_3$

Order 56: $D_{28}$ x 5, $C_2\times D_{14}$ x 5, $C_7:D_4$ x 4, $C_7\times D_4$ x 2, $C_4\times D_7$ x 2, $C_{56}$ x 2, $C_7\times Q_8$ x 2, $C_2\times C_{28}$

Order 49: $C_7^2$

Order 48: $C_2.S_4$ x 8, $C_6:D_4$ x 6, $C_2\times \SL(2,3)$ x 6, $D_4:S_3$ x 4, $S_3\times D_4$ x 4, $C_3\times Q_{16}$ x 3, $D_{24}$ x 2, $C_6\times D_4$ x 2, $D_{12}:C_2$ x 2, $\SL(2,3):C_2$ x 2, $\GL(2,3)$ x 2, $C_3\times D_8$ x 2, $C_3:D_8$ x 2, $C_6:C_8$, $C_6\times Q_8$, $C_2\times D_{12}$, $C_2\times C_{24}$, $C_{48}$, $C_{12}:C_4$, $C_{12}:C_4$, $C_3:C_{16}$

Order 42: $F_7$ x 14, $C_7:C_6$ x 12, $C_{42}$ x 3, $D_{21}$ x 2, $C_3\times D_7$ x 2

Order 36: $C_6\times S_3$ x 6, $C_3:C_{12}$ x 4, $C_6^2$ x 3, $C_3^2:C_4$ x 2, $C_6:S_3$ x 2, $S_3^2$

Order 32: $\SD_{32}$ x 14, $D_8:C_2$ x 6, $C_2\times D_8$ x 6, $C_2\times Q_{16}$ x 5, $C_2\times C_{16}$ x 5, $Q_{16}:C_2$ x 4, $D_8:C_2$ x 4, $\OD_{32}$ x 4, $Q_{32}$ x 3, $D_{16}$ x 3, $C_4.D_4$ x 2, $C_4.C_2^3$ x 2, $\OD_{16}:C_2$ x 2, $C_4:Q_8$ x 2, $C_4\times C_8$ x 2, $C_8.C_4$ x 2, $C_8:C_4$ x 2, $C_4\wr C_2$ x 2, $\OD_{16}:C_2$, $D_4:C_2^2$, $C_4:D_4$, $C_4\times Q_8$, $C_4:C_8$, $Q_8:C_4$

Order 28: $D_{14}$ x 18, $C_2\times C_{14}$ x 5, $C_{28}$ x 4, $C_7:C_4$ x 2

Order 24: $C_3:D_4$ x 16, $C_6:C_4$ x 6, $C_3\times D_4$ x 6, $C_2\times D_6$ x 5, $D_{12}$ x 4, $C_4\times S_3$ x 4, $\SL(2,3)$ x 4, $C_2^2\times C_6$ x 3, $C_2\times C_{12}$ x 2, $C_3:Q_8$ x 2, $C_{24}$ x 2, $C_3\times Q_8$ x 2, $C_3:C_8$

Order 21: $C_7:C_3$ x 4, $C_{21}$

Order 18: $C_3\times C_6$ x 3, $C_3\times S_3$ x 3, $C_3:S_3$

Order 16: $Q_{16}$ x 11, $D_8$ x 11, $\SD_{16}$ x 6, $D_4:C_2$ x 6, $C_{16}$ x 6, $C_2\times C_8$ x 5, $C_2\times Q_8$ x 5, $C_2\times D_4$ x 5, $\OD_{16}$ x 4, $C_4:C_4$, $C_4^2$

Order 14: $D_7$ x 10, $C_{14}$ x 9

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 451584: $\SL(2,7)^2.C_2^2$ ($C_1$)

Order 225792: $\SL(2,7)\wr C_2$ ($C_2$) x 2, $C_2.\SL(2,7).\SO(3,7)$ ($C_2$)

Order 112896: $\SL(2,7)^2$ ($C_2^2$)

Order 4: $C_2^2$ ($\POPlus(4,7)$)

Order 2: $C_2$ ($\GOrthPlus(4,7)$)

Order 1: $C_1$ ($\SL(2,7)^2.C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

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 $94 \times 94$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $65 \times 65$ rational character table.