Abstract group 225792.a: $\GOrthPlus(4,7)$

• Label: 225792.a
• Order: \( 2^{9} \cdot 3^{2} \cdot 7^{2} \)
• Exponent: \( 2^{4} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \cdot 7^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $128$
• Trans deg.: $128$
• Rank: $2$

Group information

Description:	$\GOrthPlus(4,7)$
Order:	\(225792\)\(\medspace = 2^{9} \cdot 3^{2} \cdot 7^{2} \)
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 3, $\PSL(2,7)$ x 2
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	8	12	14	16	21	24	28	42	48	56
Elements	1	3123	3248	17724	53424	2400	46032	4704	34656	9408	5376	9408	4032	5376	18816	8064	225792
Conjugacy classes	1	7	2	5	9	3	13	1	7	2	1	2	1	1	4	2	61
Divisions	1	7	2	5	9	3	7	1	7	1	1	1	1	1	1	1	49
Autjugacy classes	1	4	2	4	5	2	7	1	3	2	1	2	1	1	2	2	40

Dimension	1	12	14	16	18	24	32	36	49	64	72	84	96	112	128	144	168	192	384
Irr. complex chars.	4	3	2	2	4	0	4	12	4	8	3	3	9	2	1	0	0	0	0	61
Irr. rational chars.	4	1	2	2	4	1	4	4	4	8	5	1	1	2	1	1	1	2	1	49

Minimal presentations

Permutation degree:	$128$
Transitive degree:	$128$
Rank:	$2$
Inequivalent generating pairs:	$20565$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	32	32	32
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\GOPlus(4,7)$
Permutation group:	Degree $128$ $\langle(2,10)(3,15)(5,23)(6,29)(7,36)(8,41)(9,46)(11,52)(12,50)(13,22)(14,60)(17,25) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\SOPlus(4,7)$ $\,\rtimes\,$ $C_2$	$\OmegaPlus(4,7)$ $\,\rtimes\,$ $C_2^2$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $\POPlus(4,7)$				more information
Aut. group:	$\Aut(\SL(2,7)\wr C_2)$

Elements of the group are displayed as matrices in $\GOPlus(4,7)$.

Homology

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

Subgroups

There are 1250485 subgroups in 791 conjugacy classes, 7 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\POPlus(4,7)$
Commutator:	$G' \simeq$ $\OmegaPlus(4,7)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $\POPlus(4,7)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2$	$G/\operatorname{Fit} \simeq$ $\POPlus(4,7)$
Radical:	$R \simeq$ $C_2$	$G/R \simeq$ $\POPlus(4,7)$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $\POPlus(4,7)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_8\wr C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 225792: $\GOrthPlus(4,7)$

Order 112896: $C_2.\PSL(2,7)^2.C_2$ x 2, $\SOPlus(4,7)$

Order 56448: $\OmegaPlus(4,7)$

Order 14112: $C_7:\GL(2,7)$

Order 8064: $C_2.S_4.\PSL(2,7)$

Order 7056: $C_7:C_3\times \SL(2,7)$

Order 5376: $\SL(2,7):D_8$

Order 4704: $\SL(2,7):D_7$

Order 4032: $\SL(2,3).\PSL(2,7)$, $\GL(2,7):C_2$

Order 3528: $C_7^2:(C_6\times D_6)$

Order 2688: $\SL(2,7):D_4$, $Q_{16}.\PSL(2,7)$, $\Unitary(2,7)$

Order 2352: $C_7\times \SL(2,7)$

Order 2304: $\OmegaPlus(4,3):D_4$ x 2

Order 2016: $\SL(2,7):S_3$, $\SL(2,7):S_3$, $\GL(2,7)$

Order 1764: $C_7^2:(C_6\times S_3)$ x 4, $C_7^2:(C_6\times S_3)$ x 2, $C_7^2:C_6^2$

Order 1344: $C_2^2\times \PGL(2,7)$ x 2, $\SL(2,7):C_4$, $\SL(2,7):C_2^2$, $\SL(2,7):C_2^2$

Order 1176: $D_{14}:F_7$, $C_2\times C_7^2:D_6$

Order 1152: $\GOrthPlus(4,3)$ x 2, $Q_8.A_4.D_6$ x 2, $\OmegaPlus(4,3):C_4$ x 2

Order 1008: $C_{14}.(C_3\times S_4)$, $C_3\times \SL(2,7)$

Order 882: $C_7^2:(C_3\times S_3)$ x 4, $C_7:(C_3\times F_7)$ x 2, $C_2\times C_7^2:C_3^2$

Order 672: $C_2\times \PGL(2,7)$ x 12, $C_2^2\times \GL(3,2)$ x 2, $Q_{16}:F_7$, $\SL(2,7):C_2$, $\SL(2,7):C_2$

Order 588: $D_7:F_7$ x 4, $C_7^2:D_6$ x 4, $C_{14}:F_7$ x 2, $C_7^2:D_6$ x 2, $C_{14}:F_7$, $C_{14}:F_7$, $C_{14}:F_7$

Order 576: $\SOPlus(4,3)$ x 2, $\OmegaPlus(4,3):C_2$ x 2, $\SL(2,3).S_4$ x 2

Order 512: $D_8\wr C_2$

Order 504: $C_7:C_3\times \SL(2,3)$, $D_{42}:C_6$

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

Order 392: $D_7\times D_{14}$

Order 384: $C_2\wr S_4$ x 2, $\GL(2,3):D_4$

Order 336: $\PGL(2,7)$ x 8, $C_2\times \GL(3,2)$ x 6, $C_{56}.C_6$, $C_{14}.S_4$, $\SL(2,7)$, $C_{56}:C_6$, $C_7:C_{48}$

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

Order 288: $\OmegaPlus(4,3)$ x 2, $D_6\wr C_2$, $Q_8.S_3^2$

Order 256: $D_4^2:C_2^2$ x 2, $D_4^2:C_4$ x 2, $D_8:D_8$, $C_8^2:C_4$, $D_8^2$

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

Order 224: $Q_{16}:D_7$

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

Order 192: $\SL(2,3):D_4$ x 2, $C_2\wr A_4$ x 2, $Q_8:S_4$ x 2, $C_2^3:S_4$ x 2, $\GL(2,3):C_2^2$ x 2, $C_8.S_4$, $\GL(2,3):C_4$, $C_{16}:D_6$, $Q_{16}.A_4$

Order 168: $C_2^2\times F_7$ x 2, $\PSL(2,7)$ x 2, $C_{28}:C_6$, $C_7:\SL(2,3)$, $C_7\times \SL(2,3)$, $C_{28}.C_6$, $C_7:C_{24}$, $C_3:D_{28}$

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

Order 144: $S_3^2:C_2^2$ x 2, $S_3^2:C_4$ x 2, $D_6^2$, $D_6:D_6$, $C_6^2:C_4$, $\SL(2,3):S_3$, $C_3:\GL(2,3)$, $C_6.S_4$

Order 128: $D_4:D_8$ x 3, $C_2\wr D_4$ x 2, $C_8:D_8$ x 2, $D_4\times D_8$ x 2, $(C_4\times C_8):C_4$ x 2, $C_4^2.D_4$ x 2, $D_8:D_4$, $C_8.D_8$, $C_8\wr C_2$, $C_8:D_8$, $C_8\times D_8$, $D_8:D_4$

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

Order 112: $D_{56}$, $C_7\times Q_{16}$, $C_7:C_{16}$

Order 98: $C_7\times D_7$ x 4, $C_7:D_7$ x 2, $C_7\times C_{14}$

Order 96: $C_2^2\times S_4$ x 2, $Q_8:A_4$ x 2, $\SL(2,3):C_2^2$ x 2, $\GL(2,3):C_2$ x 2, $C_8.A_4$, $\Unitary(2,3)$, $C_3\times \SD_{32}$, $D_{48}$, $C_{48}:C_2$, $Q_{16}:S_3$, $C_3:D_{16}$, $\GL(2,3):C_2$, $D_{24}:C_2$, $S_3\times D_8$

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

Order 72: $S_3\times D_6$ x 8, $\SOPlus(4,2)$ x 4, $C_2\times C_3^2:C_4$ x 2, $C_6:D_6$, $C_6\times D_6$, $C_6\wr C_2$, $C_3\times \SL(2,3)$, $C_3:D_{12}$, $C_6.D_6$

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

Order 63: $C_{21}:C_3$

Order 56: $C_2\times D_{14}$ x 2, $D_{28}$, $C_{56}$, $C_7\times Q_8$

Order 49: $C_7^2$

Order 48: $C_2\times S_4$ x 12, $D_{24}$ x 2, $C_2^2\times A_4$ x 2, $\GL(2,3)$ x 2, $C_2^2\times D_6$, $D_{12}:C_2$, $S_3\times C_8$, $S_3\times D_4$, $\SL(2,3):C_2$, $C_2.S_4$, $C_3\times Q_{16}$, $C_3\times D_8$, $C_{48}$, $Q_8:S_3$, $C_3:D_8$, $C_3:C_{16}$

Order 42: $F_7$ x 11, $C_7:C_6$ x 8, $C_{42}$, $D_{21}$, $C_3\times D_7$

Order 36: $S_3^2$ x 12, $C_6\times S_3$ x 7, $C_6:S_3$ x 4, $C_3^2:C_4$ x 2, $C_3:C_{12}$, $C_6^2$

Order 32: $C_2\times D_8$ x 19, $D_4:C_4$ x 6, $C_2^2\times D_4$ x 4, $C_4:D_4$ x 4, $C_4:D_4$ x 3, $C_2^2\wr C_2$ x 3, $\OD_{16}:C_2$ x 2, $C_2^3:C_4$ x 2, $C_2^2:C_8$ x 2, $D_4:C_2^2$ x 2, $D_8:C_2$ x 2, $C_2^2\times C_8$ x 2, $C_4\times C_8$ x 2, $C_4\times D_4$ x 2, $C_4:C_8$ x 2, $D_8:C_2$, $\OD_{16}:C_2$, $\SD_{32}$, $D_{16}$, $\OD_{32}$, $C_8.C_4$, $C_8:C_4$, $C_4\wr C_2$

Order 28: $D_{14}$ x 13, $C_2\times C_{14}$ x 2, $C_{28}$

Order 24: $C_2\times D_6$ x 15, $S_4$ x 8, $C_2\times A_4$ x 6, $D_{12}$ x 2, $\SL(2,3)$ x 2, $C_3:D_4$, $C_4\times S_3$, $C_{24}$, $C_2^2\times C_6$, $C_3\times Q_8$, $C_3\times D_4$, $C_3:C_8$

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

Order 18: $C_3\times S_3$ x 5, $C_3:S_3$ x 3, $C_3\times C_6$ x 2

Order 16: $C_2\times D_4$ x 25, $D_8$ x 22, $C_2\times C_8$ x 9, $C_2^2:C_4$ x 5, $C_2^4$ x 3, $C_4:C_4$ x 2, $C_2^2\times C_4$ x 2, $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $C_4^2$, $D_4:C_2$, $C_{16}$

Order 14: $D_7$ x 9, $C_{14}$ x 7

Order 12: $D_6$ x 32, $C_2\times C_6$ x 8, $A_4$ x 2, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $D_4$ x 24, $C_2^3$ x 14, $C_2\times C_4$ x 8, $C_8$ x 7, $Q_8$

Order 7: $C_7$ x 3

Order 6: $S_3$ x 10, $C_6$ x 9

Order 4: $C_2^2$ x 18, $C_4$ x 5

Order 3: $C_3$ x 2

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 225792: $\GOrthPlus(4,7)$

Order 112896: $C_2.\PSL(2,7)^2.C_2$, $\SOPlus(4,7)$

Order 56448: $\OmegaPlus(4,7)$

Order 14112: $C_7:\GL(2,7)$

Order 8064: $C_2.S_4.\PSL(2,7)$

Order 7056: $C_7:C_3\times \SL(2,7)$

Order 5376: $\SL(2,7):D_8$

Order 4704: $\SL(2,7):D_7$

Order 4032: $\SL(2,3).\PSL(2,7)$, $\GL(2,7):C_2$

Order 3528: $C_7^2:(C_6\times D_6)$

Order 2688: $\SL(2,7):D_4$, $Q_{16}.\PSL(2,7)$, $\Unitary(2,7)$

Order 2352: $C_7\times \SL(2,7)$

Order 2304: $\OmegaPlus(4,3):D_4$

Order 2016: $\SL(2,7):S_3$, $\SL(2,7):S_3$, $\GL(2,7)$

Order 1764: $C_7^2:C_6^2$, $C_7^2:(C_6\times S_3)$, $C_7^2:(C_6\times S_3)$

Order 1344: $\SL(2,7):C_4$, $C_2^2\times \PGL(2,7)$, $\SL(2,7):C_2^2$, $\SL(2,7):C_2^2$

Order 1176: $D_{14}:F_7$, $C_2\times C_7^2:D_6$

Order 1152: $\GOrthPlus(4,3)$, $Q_8.A_4.D_6$, $\OmegaPlus(4,3):C_4$

Order 1008: $C_{14}.(C_3\times S_4)$, $C_3\times \SL(2,7)$

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

Order 672: $C_2\times \PGL(2,7)$ x 3, $Q_{16}:F_7$, $C_2^2\times \GL(3,2)$, $\SL(2,7):C_2$, $\SL(2,7):C_2$

Order 588: $C_{14}:F_7$, $C_{14}:F_7$, $C_{14}:F_7$, $C_{14}:F_7$, $C_7^2:D_6$, $D_7:F_7$, $C_7^2:D_6$

Order 576: $\SOPlus(4,3)$, $\OmegaPlus(4,3):C_2$, $\SL(2,3).S_4$

Order 512: $D_8\wr C_2$

Order 504: $C_7:C_3\times \SL(2,3)$, $D_{42}:C_6$

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

Order 392: $D_7\times D_{14}$

Order 384: $C_2\wr S_4$, $\GL(2,3):D_4$

Order 336: $C_2\times \GL(3,2)$ x 2, $\PGL(2,7)$ x 2, $C_{56}.C_6$, $C_{14}.S_4$, $\SL(2,7)$, $C_{56}:C_6$, $C_7:C_{48}$

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

Order 288: $D_6\wr C_2$, $\OmegaPlus(4,3)$, $Q_8.S_3^2$

Order 256: $D_4^2:C_2^2$, $D_8:D_8$, $C_8^2:C_4$, $D_4^2:C_4$, $D_8^2$

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

Order 224: $Q_{16}:D_7$

Order 196: $D_7^2$, $C_7:D_{14}$, $C_7\times D_{14}$

Order 192: $\SL(2,3):D_4$, $C_8.S_4$, $\GL(2,3):C_4$, $C_{16}:D_6$, $C_2\wr A_4$, $Q_8:S_4$, $C_2^3:S_4$, $\GL(2,3):C_2^2$, $Q_{16}.A_4$

Order 168: $C_{28}:C_6$, $C_2^2\times F_7$, $\PSL(2,7)$, $C_7:\SL(2,3)$, $C_7\times \SL(2,3)$, $C_{28}.C_6$, $C_7:C_{24}$, $C_3:D_{28}$

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

Order 144: $D_6^2$, $S_3^2:C_2^2$, $D_6:D_6$, $C_6^2:C_4$, $\SL(2,3):S_3$, $C_3:\GL(2,3)$, $C_6.S_4$, $S_3^2:C_4$

Order 128: $D_4:D_8$ x 2, $D_8:D_4$, $C_2\wr D_4$, $C_8.D_8$, $C_8\wr C_2$, $C_8:D_8$, $C_8:D_8$, $C_8\times D_8$, $D_8:D_4$, $D_4\times D_8$, $(C_4\times C_8):C_4$, $C_4^2.D_4$

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

Order 112: $D_{56}$, $C_7\times Q_{16}$, $C_7:C_{16}$

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

Order 96: $C_8.A_4$, $\Unitary(2,3)$, $C_3\times \SD_{32}$, $D_{48}$, $C_{48}:C_2$, $Q_{16}:S_3$, $C_3:D_{16}$, $C_2^2\times S_4$, $Q_8:A_4$, $\SL(2,3):C_2^2$, $\GL(2,3):C_2$, $\GL(2,3):C_2$, $D_{24}:C_2$, $S_3\times D_8$

Order 84: $C_2\times F_7$ x 5, $C_{14}:C_6$, $C_3:C_{28}$, $C_7:C_{12}$, $D_{42}$, $C_3\times D_{14}$

Order 72: $S_3\times D_6$ x 4, $C_6:D_6$, $C_6\times D_6$, $C_2\times C_3^2:C_4$, $\SOPlus(4,2)$, $C_6\wr C_2$, $C_3\times \SL(2,3)$, $C_3:D_{12}$, $C_6.D_6$

Order 64: $C_4:D_8$ x 3, $D_4^2$ x 2, $C_4:D_8$ x 2, $D_4:D_4$ x 2, $C_4.D_8$ x 2, $D_4:C_8$, $C_8.C_8$, $\OD_{32}:C_2$, $C_4^2:C_4$, $C_2\wr C_4$, $D_8:C_2^2$, $C_2^2\times D_8$, $C_8^2$, $D_{16}:C_2$, $C_8:C_8$, $C_8.D_4$, $C_8:D_4$, $C_2\wr C_2^2$, $D_4:D_4$, $D_8:C_4$, $C_4\times D_8$, $C_8\times D_4$

Order 63: $C_{21}:C_3$

Order 56: $D_{28}$, $C_{56}$, $C_2\times D_{14}$, $C_7\times Q_8$

Order 49: $C_7^2$

Order 48: $C_2\times S_4$ x 4, $D_{24}$ x 2, $\GL(2,3)$ x 2, $C_2^2\times D_6$, $C_2^2\times A_4$, $D_{12}:C_2$, $S_3\times C_8$, $S_3\times D_4$, $\SL(2,3):C_2$, $C_2.S_4$, $C_3\times Q_{16}$, $C_3\times D_8$, $C_{48}$, $Q_8:S_3$, $C_3:D_8$, $C_3:C_{16}$

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

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

Order 32: $C_2\times D_8$ x 8, $D_4:C_4$ x 3, $C_2^2\times D_4$ x 2, $C_4:D_4$ x 2, $C_4\times C_8$ x 2, $C_4:D_4$ x 2, $C_2^2\wr C_2$ x 2, $\OD_{16}:C_2$, $C_2^3:C_4$, $C_2^2:C_8$, $D_4:C_2^2$, $D_8:C_2$, $D_8:C_2$, $\OD_{16}:C_2$, $C_2^2\times C_8$, $C_4\times D_4$, $\SD_{32}$, $D_{16}$, $\OD_{32}$, $C_8.C_4$, $C_8:C_4$, $C_4:C_8$, $C_4\wr C_2$

Order 28: $D_{14}$ x 4, $C_2\times C_{14}$, $C_{28}$

Order 24: $C_2\times D_6$ x 6, $S_4$ x 3, $D_{12}$ x 2, $\SL(2,3)$ x 2, $C_2\times A_4$ x 2, $C_3:D_4$, $C_4\times S_3$, $C_{24}$, $C_2^2\times C_6$, $C_3\times Q_8$, $C_3\times D_4$, $C_3:C_8$

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

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

Order 16: $C_2\times D_4$ x 11, $D_8$ x 9, $C_2\times C_8$ x 5, $C_2^2:C_4$ x 3, $C_2^4$ x 2, $Q_{16}$, $\SD_{16}$, $\OD_{16}$, $C_4:C_4$, $C_4^2$, $D_4:C_2$, $C_2^2\times C_4$, $C_{16}$

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

Order 12: $D_6$ x 11, $C_2\times C_6$ x 4, $A_4$, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $D_4$ x 10, $C_2^3$ x 6, $C_2\times C_4$ x 5, $C_8$ x 5, $Q_8$

Order 7: $C_7$ x 2

Order 6: $C_6$ x 5, $S_3$ x 5

Order 4: $C_2^2$ x 8, $C_4$ x 4

Order 3: $C_3$ x 2

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 225792: $\GOrthPlus(4,7)$ ($C_1$)

Order 112896: $C_2.\PSL(2,7)^2.C_2$ ($C_2$) x 2, $\SOPlus(4,7)$ ($C_2$)

Order 56448: $\OmegaPlus(4,7)$ ($C_2^2$)

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

Order 1: $C_1$ ($\GOrthPlus(4,7)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 225792: $\GOrthPlus(4,7)$ ($C_1$)

Order 112896: $C_2.\PSL(2,7)^2.C_2$ ($C_2$), $\SOPlus(4,7)$ ($C_2$)

Order 56448: $\OmegaPlus(4,7)$ ($C_2^2$)

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

Order 1: $C_1$ ($\GOrthPlus(4,7)$)

Series

Derived series	$\GOrthPlus(4,7)$	$\rhd$	$\OmegaPlus(4,7)$
Chief series	$\GOrthPlus(4,7)$	$\rhd$	$\SOPlus(4,7)$	$\rhd$	$\OmegaPlus(4,7)$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$\GOrthPlus(4,7)$	$\rhd$	$\OmegaPlus(4,7)$
Upper central series	$C_1$	$\lhd$	$C_2$

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 $61 \times 61$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $49 \times 49$ rational character table.