Abstract group $\SO(7,3)$

• Label: 9170703360.b
• Order: \( 2^{10} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\mathrm{Aut}(G)}$: \( 2^{10} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $351$
• Trans deg.: $351$
• Rank: not computed

Group information

Description:	$\SO(7,3)$
Order:	\(9170703360\)\(\medspace = 2^{10} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 \)
Exponent:	\(32760\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \)
Automorphism group:	$\SO(7,3)$, of order \(9170703360\)\(\medspace = 2^{10} \cdot 3^{9} \cdot 5 \cdot 7 \cdot 13 \)
Outer automorphisms:	$C_1$, of order $1$
Composition factors:	$C_2$, $B(3,3)$
Derived length:	$1$

This group is nonabelian, almost simple, and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	12	13	14	15	18	20	24	26	28	30	36
Elements	1	664119	5307848	79632072	38211264	535488408	327525120	764225280	382112640	726014016	1489531680	705438720	327525120	305690112	721768320	458535168	382112640	705438720	655050240	305690112	254741760	9170703360
Conjugacy classes	1	6	6	9	1	26	1	4	3	5	17	2	1	1	6	2	1	2	2	1	1	98
Divisions	1	6	6	9	1	26	1	4	3	5	17	1	1	1	6	2	1	1	1	1	1	95
Autjugacy classes	1	6	6	9	1	26	1	4	3	5	17	2	1	1	6	2	1	2	2	1	1	98

Dimension	1	78	91	105	168	182	195	273	520	546	819	1092	1365	1638	1820	2106	2184	2457	2730	2835	3120	4095	4368	4536	5265	5460	5824	6552	7280	7371	11648	14040	14560	14742	16380	16640	17472	17920	19683	21840	22113	33280	35840
Irr. complex chars.	2	2	2	2	2	2	2	2	1	2	2	2	4	2	3	2	2	2	4	2	1	4	2	2	2	6	4	2	2	2	2	1	1	2	3	4	4	4	2	2	2	0	0	98
Irr. rational chars.	2	2	2	2	2	2	2	2	1	2	2	2	4	2	3	2	2	2	4	2	1	4	2	2	2	6	4	2	2	2	2	1	1	2	3	2	4	0	2	2	2	1	2	95

Minimal Presentations

Permutation degree:	$351$
Transitive degree:	$351$
Rank:	not computed
Inequivalent generating tuples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	78	78	78
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\SO(7,3)$, $\PGO(7,3)$
Permutation group:	Degree $351$ $\langle(1,2,4)(3,6,11,20,36,60)(5,9,16,28,8,14)(7,13)(10,18,32,17,30,51)(12,22,39,21,37,62) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$B(3,3)$ . $C_2$				more information
Aut. group:	$\Aut(B(3,3))$

Elements of the group are displayed as matrices in $\SO(7,3)$.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 3 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $B(3,3)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4.C_3^4.C_3$

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 subgroup of index up to 580608 (order at least 15795) are shown, as well as all normal subgroups of any index.

Order 9170703360: $\SO(7,3)$

Order 4585351680: $B(3,3)$

Order 26127360: $C_2.2A(3,3).C_2^2$

Order 25194240: $C_3^2:(C_3^3:C_2).C(2,3).C_2$

Order 24261120: $\GOPlus(6,3)$

Order 13063680: $C_2.2A(3,3).C_2$ x 2, $\SOMinus(6,3)$

Order 12597120: $C_3^5.C(2,3).C_2$ x 2, $C_3^2:(C_3^3:C_2).C(2,3)$

Order 12130560: $\PGOPlus(6,3)$ x 2, $C_2\times \PSL(4,3)$

Order 8188128: $C_3^3.C_3^3.C_2.{\rm PSL}(3,3)$

Order 6531840: $\OmegaMinus(6,3)$

Order 6298560: $C_3^5.C(2,3)$

Order 6065280: $\PSL(4,3)$

Order 4245696: $G(2,3)$

Order 4094064: $C_3^3.C_3^3.{\rm PSL}(3,3)$

Order 2519424: $C_3.C_3^6.Q_8.A_4.D_6$

Order 1451520: $\Sp(6,2)$

Order 1259712: $C_3.C_3^6.C_2.A_4^2.C_2$ x 2, $C_3.C_3^6.Q_8.A_4.S_3$

Order 933120: $S_3^5.S_5$

Order 839808: $C_3^3:C_3^2.C_3:S_3.(D_4\times S_4)$, $C_3.C_3^6.(D_4\times S_4).C_2$

Order 699840: $C_3^5.C_2^2.A_6.C_2$

Order 629856: $C_3^4.C_3^4.Q_8.D_6$, $C_3^4.C_3^3.(C_6\times S_4).C_2$, $C_3^3.C_3^4.Q_8.S_3^2$, $C_3.C_3^6.C_2.A_4^2$

Order 559872: $C_3^4.Q_8:C_2^2.S_3^3$

Order 466560: $C_3:S_3^4:S_5$, $S_3\wr A_5$, $C_3:S_3^4:S_5$

Order 419904: $He_3.C_3^4.Q_8.C_6.C_2^2$ x 3, $C_3.C_3^6.A_4.{\rm SD}_{16}$ x 3, $He_3.C_3^4.Q_8.C_3.D_4$, $He_3.C_3^2.Q_8.C_3.D_4$, $C_3^3:C_3^2.C_3:S_3.A_4.D_4$, $C_3^3:C_3^2.C_3:S_3.(C_4\times S_4)$, $C_3.C_3^6.C_2^3.S_4$, $C_3.C_3^6.(Q_8\times S_4)$, $C_3.C_3^6.(D_4\times S_4)$, $C_3.C_3^6.(C_8\times S_4)$, $C_3.C_3^6.(A_4\times {\rm SD}_{16})$

Order 414720: $C(2,3).C_2\times D_4$

Order 362880: $S_9$

Order 349920: $C_3^2:(C_3^3:C_2).A_6.C_2$ x 2, $(C_3^4\times S_3).A_6.C_2$ x 2, $(C_3\times C_3:(C_3^3:C_2)).A_6.C_2$ x 2, $C_3^4.D_6.A_6$

Order 322560: $C_2^6.S_7$

Order 314928: $C_3^4.C_3^3.A_4.D_6$ x 3, $C_3^4.C_3^4.Q_8.S_3$ x 2, $C_3^4.C_3^4.Q_8.C_6$, $C_3^4.C_3^4.A_4.C_2^2$, $C_3^4.C_3^3.C_3:S_4.C_2$, $C_3^4.C_3^3.(S_3\times S_4)$, $C_3^4.C_3^3.(C_6\times S_4)$, $C_3^3.C_3^4.Q_8.C_3.S_3$, $C_3^3.C_3^4.Q_8.C_3.C_6$, $C_3^3.C_3^4.C_6.A_4.C_2$

Order 303264: $C_2\times C_3^3.{\rm PSL}(3,3)$

Order 279936: $C_3^4.Q_8:C_2^2.C_3^3.C_2^2$ x 2, $C_3^4.Q_8:C_2^2.C_3^2.D_6$ x 2, $He_3.C_3^2.D_4^2.C_2$, $C_3^4.Q_8.(C_3\times A_4).D_6$, $C_3^4.C_2^3:A_4.S_3^2$, $C_3^4.C_2^3:A_4.(C_6\times S_3)$

Order 233280: $(C_2\times C_3:(C_3^3:C_2)).A_6.C_2$, $C_3:S_3^4:A_5$

Order 209952: $He_3.C_3^4.Q_8.D_6$ x 5, $He_3.C_3^4.Q_8.C_6.C_2$ x 2, $C_3.C_3^6.A_4.D_4$ x 2, $C_3.C_3^6.A_4.C_8$ x 2, $He_3.C_3^4.Q_8.C_{12}$, $He_3.C_3^2.Q_8.C_6.C_2$, $C_3^4.C_3^4.D_4.C_2^2$, $C_3^4.C_3^4.C_4.C_2^3$, $C_3^4.C_3^3.S_4.C_2^2$, $C_3^3.C_3^4.C_{12}.C_2^3$, $C_3.C_3^6.C_2^3.A_4$, $C_3.C_3^6.A_4.Q_8$, $C_3.C_3^6.(Q_8\times A_4)$, $C_3.C_3^6.(D_4\times A_4)$, $C_3.C_3^6.(C_4\times S_4)$

Order 207360: $C_2^2\times C(2,3).C_2$ x 3, $C_2^2.C(2,3).C_2$ x 2, $C_4\times C(2,3).C_2$, $C_4.C(2,3).C_2$, $C(2,3)\times D_4$

Order 186624: $S_3^5.S_4$ x 2, $C_3^4.C_2^3:A_4.D_6.C_2$

Order 181440: $A_9$

Order 174960: $C_3^5.A_6.C_2$ x 4, $C_3^2:(C_3^3:C_2).A_6$, $(C_3^4\times S_3).A_6$, $(C_3\times C_3:(C_3^3:C_2)).A_6$

Order 161280: $C_2^6:A_7$

Order 157464: $C_3^4.C_3^4.A_4.C_2$ x 2, $C_3^4.C_3^3.A_4.C_6$ x 2, $C_3^4.C_3^4.Q_8.C_3$, $C_3^4.C_3^3.C_3:S_4$, $C_3^4.C_3^3.A_4.S_3$, $C_3^4.C_3^3.(C_6\times A_4)$, $C_3^3.C_3^4.Q_8.C_3^2$, $C_3.C_3^5.S_3^3$

Order 155520: $S_3\wr F_5$

Order 151632: $C_3^3.\SL(3,3)$

Order 139968: $C_3^4.C_2^3:A_4.C_3.C_6$ x 4, $He_3.C_3^4.C_4.C_2^4$ x 2, $He_3.C_3^4.D_4^2$, $He_3.C_3^4.C_4^2.C_2^2$, $He_3.C_3^4.C_4:{\rm SD}_{16}$, $He_3.C_3^4.C_2.C_2^4.C_2$, $He_3.C_3^2.C_4^2.C_2^2$, $He_3.C_3^2.C_4:D_4.C_2$, $He_3.C_3^2.C_2^2:{\rm SD}_{16}$, $He_3.C_3^2.C_2^2:D_4.C_2$, $C_3^4.Q_8.C_3^2:S_4$, $C_3^4.Q_8.A_4.C_3.C_6$, $C_3^4.(C_3\times Q_8.A_4).C_6$, $C_3^3:C_3^2.C_3^4.(D_4\times Q_8)$, $C_3^3:C_3^2.C_3^2.(C_8\times D_4)$, $C_3^3:C_3^2.C_3:S_3.C_2^4.C_2$, $C_3^3:C_3^2.C_3:S_3.C_2^2:D_4$, $C_3^3:C_3^2.C_2^2:S_3{\rm wrC}_2.C_2$

Order 116640: $C_3^4:(C_2\times S_6)$ x 4, $(C_3^3\times C_6).A_6.C_2$ x 2, $C_3^4:(D_6\times S_5)$ x 2, $(C_2\times C_3:(C_3^3:C_2)).A_6$

Order 104976: $C_3^4.C_3^4.{\rm SD}_{16}$ x 4, $C_3^4.C_3^4.C_2^3.C_2$ x 4, $C_3^4.C_3^3.A_4.C_2^2$ x 4, $He_3.C_3^4.Q_8.S_3$ x 3, $C_3^3.C_3^4.C_6.D_4$ x 3, $He_3.C_3^4.Q_8.C_6$ x 2, $C_3^4.C_3^4.D_4.C_2$ x 2, $C_3^4.C_3^3.S_4.C_2$ x 2, $C_3^3.C_3^4.Q_8.S_3$ x 2, $C_3^3.C_3^4.D_6.C_2^2$ x 2, $C_3^3.C_3^4.C_6.C_2^3$ x 2, $C_3.C_3^6.A_4.C_4$ x 2, $C_3.C_3^5.D_6:D_6$ x 2, $C_3.C_3^5.C_6^2.C_2^2$ x 2, $C_3.C_3^5.(S_3\times D_{12})$ x 2, $C_3^4.C_3^4.C_2^4$, $C_3^4.C_3^4.C_2.C_2^3$, $C_3^3.{\rm He}_3.(C_2\times F_9)$, $C_3^3.C_3^4.C_4.C_6.C_2$, $C_3^3.C_3^4.(C_8\times S_3)$, $C_3.C_3^5.D_6^2$

Order 103680: $C_2\times \SO(5,3)$ x 13, $C_2^2\times C(2,3)$ x 3, $C_4\times C(2,3)$, $C_2.C(2,3).C_2$

Order 93312: $C_3^4:\SOPlus(4,3):C_2$ x 6, $C_3^4.C_2^3:A_4.C_6.C_2$ x 2, $S_3\times S_3\wr A_4$ x 2, $C_3:S_3\wr S_4$ x 2, $C_3:S_3\wr S_4$ x 2, $C_3:S_3\wr S_4$ x 2, $C_3\times S_3\wr S_4$ x 2, $C_3^5:(C_2^4.S_4)$ x 2, $C_3:S_3^4:S_4$ x 2, $He_3:(C_3:S_3).(D_4\times S_4)$, $C_3^4.Q_8.C_3.D_6.C_2^2$, $C_3^4.Q_8.A_4.D_6$, $S_3^5.D_6$

Order 87480: $C_3^5.A_6$

Order 80640: $C_2.{\rm PSL}(3,4).C_2$

Order 78732: $C_3^4.C_3^4.D_6$ x 2, $C_3^4.C_3^4.C_6.C_2$ x 2, $C_3^4.{\rm He}_3.S_3^2$, $C_3^4.C_3{\rm wrC}_3.D_6$, $C_3^4.C_3^4:C_6.C_2$, $C_3^4.C_3^3.A_4.C_3$

Order 77760: $S_3\wr D_5$, $C_3:S_3^4:F_5$, $C_3:S_3^4:F_5$

Order 69984: $He_3.C_3^4.C_4.C_2^3$ x 14, $He_3.C_3^4.Q_8:C_4$ x 3, $He_3.C_3^4.C_2^4.C_2$ x 2, $He_3.C_3^4.C_2^2:D_4$ x 2, $He_3.C_3^4.C_2.C_2^4$ x 2, $C_3^3:C_3^2.S_3{\rm wrC}_2:C_4$ x 2, $C_3^3:C_3^2.C_2^2:F_9$ x 2, $(C_3^2\times {\rm He}_3).D_6{\rm wrC}_2$ x 2, $He_3.C_3^4.C_4^2.C_2$, $He_3.C_3^4.C_4:Q_8$, $He_3.C_3^4.C_4:D_4$, $He_3.C_3^2.D_4:C_4$, $He_3.C_3^2.C_2^2:Q_8$, $He_3.C_3^2.C_2^2:D_4$, $C_3^4.Q_8.C_6^2.C_3$, $C_3^4.Q_8.C_3^2.D_6$, $C_3^3:C_3^2.C_4:F_9$, $C_3^3:C_3^2.C_3^4.C_2^2:Q_8$, $C_3^3:C_3^2.C_3^2.C_4^2.C_2$, $C_3^3:C_3^2.C_3^2.(C_4\times D_4)$, $C_3^3:C_3^2.(C_4\times S_3{\rm wrC}_2)$, $C_3^3.C_3^4.C_8:C_4$, $C_3^3.C_3^4.C_4.C_2^3$, $C_3^3.C_3^3.D_6.C_2^3$, $C_3.C_3^5.C_6.C_2^4$, $(C_3^2\times {\rm He}_3).Q_8.S_3^2$

Order 62208: $C_2\times S_3\wr S_4$ x 2, $S_3^5.D_4$

Order 58320: $C_3^4:S_6$ x 4, $C_3^5:(C_2\times S_5)$ x 2, $S_3\times C_3^4:S_5$ x 2, $C_3^4:(C_2\times A_6)$ x 2, $C_3^4:(S_3\times S_5)$ x 2, $C_3^4:(D_6\times A_5)$ x 2, $C_3^4:(C_6\times S_5)$ x 2, $S_3\times C_3^4:S_5$ x 2, $C_3^4:(S_3\times S_5)$ x 2, $(C_3^3\times C_6).A_6$

Order 56862: $C_3^3.C_3^3.C_{13}.C_6$

Order 52488: $C_3^4.C_3^4.C_2^3$ x 12, $C_3^4.C_3^4.D_4$ x 11, $C_3^4.C_3^3.S_4$ x 5, $C_3.C_3^5.C_6^2.C_2$ x 4, $C_3^4.C_3^3.A_4.C_2$ x 3, $C_3^3.C_3^4.C_6.C_2^2$ x 3, $C_3.C_3^5.C_6.C_6.C_2$ x 3, $C_3^5.S_3^3$ x 2, $C_3^4.C_3^4.Q_8$ x 2, $C_3^4.C_3^4.C_4.C_2$ x 2, $C_3^4.C_3^4.C_2^2.C_2$ x 2, $C_3^4.C_3^3.D_6.C_2$ x 2, $C_3^3.C_3^4.Q_8.C_3$ x 2, $C_3^3.C_3^4.C_3.D_4$ x 2, $He_3.C_3^4.Q_8.C_3$, $He_3.C_3^4.C_6.C_2^2$, $C_3^5.C_3^3.C_2^3$, $C_3^5.C_3.C_6.C_2^2$, $C_3^4.C_3^4.C_8$, $C_3^3.{\rm He}_3.F_9$, $C_3^3.C_3^4:C_3.C_2^2.C_2$, $C_3^3.C_3^4.D_{12}$, $C_3^3.C_3^4.C_{24}$, $C_3^3.C_3^4.C_{12}.C_2$, $C_3^3.C_3^4.C_3.Q_8$, $C_3^3.C_3^3.F_9$, $C_3^3.C_3^3.C_6^2.C_2$

Order 51840: $\SO(5,3)$ x 9, $C_2\times C(2,3)$ x 6

Order 48384: $C_4.2A(2,3).C_2$

Order 46656: $C_3^4:\OmegaPlus(4,3):C_2$ x 8, $C_3^4:\SOPlus(4,3)$ x 5, $He_3.{\rm ASL}(2,3).C_2^3$ x 3, $S_3^5:S_3$ x 3, $He_3:(C_3:S_3).A_4.D_4$ x 2, $C_3^5:(D_4\times S_4)$ x 2, $C_3^5:(D_4\times S_4)$ x 2, $C_3^5:(C_2\times \GL(2,\mathbb{Z}/4))$ x 2, $C_3:S_3^4:A_4$ x 2, $C_3^5:Q_8:S_4$ x 2, $C_3^5:Q_8:S_4$ x 2, $C_3^2:S_3^3:S_4$ x 2, $C_3^2:S_3^3:S_4$ x 2, $C_3\times S_3\wr A_4$ x 2, $C_3:S_3\wr A_4$ x 2, $He_3:(C_3:S_3).(D_4\times A_4)$, $C_3^4.Q_8.S_3^2.C_2$, $C_3^4.Q_8.D_6:S_3$, $C_3^4.Q_8.C_6.C_6.C_2$, $C_3^4.Q_8.C_3^2.D_4$, $C_3^4.Q_8.C_3.D_{12}$, $C_3^4.Q_8.C_3.D_6.C_2$, $C_3^4.Q_8.A_4.C_6$, $C_3^4.(C_3\times Q_8).C_6.C_2^2$, $C_3^3:C_3^2.Q_8.D_{12}$, $C_3^3:C_3^2.Q_8.D_6.C_2$, $C_3^5:(D_4\times S_4)$, $C_3^5:(D_4\times S_4)$, $C_3^5:(D_4\times S_4)$, $C_3^5:(D_4\times S_4)$, $(C_3^2\times S_3^3):D_{12}$, $C_3:S_3^4.D_6$, $S_3^5.C_6$

Order 46080: $C_2^6.S_6$ x 2

Order 40320: $C_2.\PSL(3,4)$, $S_8$

Order 39366: $C_3^4.C_3^4.C_6$ x 3, $C_3^5.C_3^3.C_6$, $C_3^4.C_3^4:C_6$, $C_3^4.C_3^4.S_3$, $C_3.C_3^5.C_3^3.C_2$

Order 38880: $C_3^4:(C_2^2\times S_5)$ x 2, $S_3\wr C_5$, $C_3:S_3^4:D_5$, $C_3:S_3^4:D_5$

Order 34992: $He_3.C_3^4.{\rm SD}_{16}$ x 11, $He_3.C_3^4.C_2^3.C_2$ x 10, $He_3.C_3^4.C_4.C_2^2$ x 6, $C_3^3.C_3^4.D_4.C_2$ x 6, $C_3^3.C_3^4.C_2^3.C_2$ x 6, $C_3.C_3^5.C_6.C_2^3$ x 5, $He_3.C_3^4.C_4:C_4$ x 4, $He_3.C_3^4.C_2.C_2^3$ x 4, $C_3.C_3^5.D_6.C_2^2$ x 4, $He_3.C_3^4.C_2^2:C_4$ x 2, $C_3^4.Q_8.C_3^3.C_2$ x 2, $C_3^3.C_3^4.{\rm SD}_{16}$ x 2, $C_3^3.C_3^4.C_2^4$ x 2, $C_3^3.C_3^3.C_6.C_2^3$ x 2, $C_3:({\rm He}_3:S_3).C_3:S_4$ x 2, $He_3.C_3^4.C_4^2$, $C_3^5.D_6^2$, $C_3^5.C_3:S_3.C_2^3$, $C_3^4.S_3^3.C_2$, $C_3^4.C_3^3.C_2^4$, $C_3^4.C_3^3.C_2.C_2^3$, $C_3^4.(C_3\times Q_8).C_3.C_6$, $C_3^3:C_3^2.D_6:D_6$, $C_3^3.{\rm ASL}(2,3).C_6$, $C_3^3.C_3^4.C_2.C_2^3$, $C_3.C_3^5.D_{12}.C_2$, $(C_3^2\times {\rm He}_3).C_6.A_4.C_2$, $C_3^3.S_3\wr S_3$

Order 34560: $C_2\times S_4\times S_6$

Order 31104: $S_3\wr S_4$ x 10, $C_2\times S_3\wr A_4$ x 2, $S_3^5:C_2^2$ x 2, $S_3\times S_3\wr C_4$ x 2, $S_3^4:D_{12}$ x 2, $C_3:S_3\wr D_4$ x 2, $C_6:S_3^3.S_4$ x 2, $C_6:S_3^3:S_4$ x 2, $He_3:(C_3:S_3).D_4^2$, $C_3^4.Q_8.C_6.C_2^3$, $S_3^5:C_2^2$, $C_3:S_3^4.D_4$, $C_3^5:(C_4^2:C_2^3)$, $C_3:S_3\wr D_4$, $C_3:S_3\wr D_4$, $\SOPlus(4,2)^2:S_3$, $C_3\times S_3\wr D_4$

Order 29160: $C_3^4:(C_6\times A_5)$ x 2, $C_3^5:S_5$ x 2, $S_3\times C_3^4:A_5$ x 2, $C_3\wr S_5$ x 2, $C_3^5:S_5$ x 2, $C_3^4:(S_3\times A_5)$ x 2, $C_3^5:S_5$ x 2, $C_3^4:A_6$

Order 28431: $C_3^3.C_3^3.C_{13}.C_3$

Order 27648: $C_2.C_2^6.S_3^3$

Order 26244: $C_3^4.C_3^3.D_6$ x 14, $C_3^3.C_3^4.C_6.C_2$ x 9, $C_3^4.C_3^4.C_4$ x 5, $C_3^4.C_3^4.C_2^2$ x 5, $C_3.C_3^5.C_6^2$ x 3, $C_3^5.C_3^3.C_2^2$ x 2, $C_3^5.C_3^2.D_6$ x 2, $C_3^4.C_3^3.C_6.C_2$ x 2, $C_3^4.C_3^3.A_4$ x 2, $C_3^5.C_3^3.C_4$, $C_3^4.{\rm He}_3.D_6$, $C_3^4.{\rm He}_3.C_6.C_2$, $C_3^4.C_9.(C_6\times S_3)$, $C_3^3.C_3^4:C_3.C_4$, $C_3^3.C_3^4.C_{12}$, $C_3^3.C_3^3.C_6^2$

Order 25920: $C(2,3)$ x 3

Order 24192: $C_2\times 2A(2,3).C_2$ x 2, $\GU(3,3)$

Order 23328: $C_3^3:C_3^2.Q_8.D_6$ x 12, $C_3^4:\OmegaPlus(4,3)$ x 4, $(C_3\times S_3^4):S_3$ x 4, $C_3\times S_3^4:S_3$ x 4, $C_3^4:(D_6\times S_4)$ x 4, $C_3^4:(D_6\times S_4)$ x 4, $He_3.{\rm ASL}(2,3).C_2^2$ x 3, $C_3^3:S_4\times S_3^2$ x 3, $C_3^4:(D_6\times S_4)$ x 3, $C_3^4:(D_6\times S_4)$ x 3, $C_3^3:S_4\times S_3^2$ x 3, $S_3\wr C_3\times S_3^2$ x 3, $C_3^4:(D_6\times S_4)$ x 3, $C_3^4.Q_8.S_3^2$ x 2, $C_3^4.Q_8.C_3.D_6$ x 2, $C_3^4:(D_6\times S_4)$ x 2, $C_3^5:C_4:S_4$ x 2, $C_3^5:(D_4\times A_4)$ x 2, $C_3^5:\GL(2,\mathbb{Z}/4)$ x 2, $C_3^5:\GL(2,\mathbb{Z}/4)$ x 2, $C_3^5:\GL(2,\mathbb{Z}/4)$ x 2, $C_3^5:\GL(2,\mathbb{Z}/4)$ x 2, $S_3\times C_3^4:\GL(2,3)$ x 2, $C_3^2:S_3^3:A_4$ x 2, $He_3:(C_3:S_3).A_4.C_4$, $He_3.{\rm PSU}(3,2).D_6$, $C_3^4.{\rm SL}(2,3):D_6$, $C_3^4.Q_8.C_6^2$, $C_3^4.Q_8.C_3^2.C_4$, $C_3^4.Q_8.C_3.C_{12}$, $C_3^4.Q_8.(C_6\times S_3)$, $C_3^4.(S_3\times Q_8:S_3)$, $C_3^4.(C_3\times Q_8).C_6.C_2$, $C_3^3:C_3^2.Q_8.C_{12}$, $C_3^3.C_3^3.C_4.C_2^3$, $C_3^3.C_3^3.C_2.C_2^4$, $C_3.C_3^5.D_4.C_2^2$, $C_2\times {\rm He}_3.{\rm AGL}(2,3)$, $C_3^4:C_{12}:S_4$, $C_3^4:C_{12}:S_4$, $C_3^5:(C_4\times S_4)$, $C_3^5:(C_4\times S_4)$, $C_3^5:(C_4\times S_4)$, $C_3^5:(C_4\times S_4)$, $C_3^5:(D_4\times A_4)$, $C_3^5:(D_4\times A_4)$, $S_3^3:C_3^3:C_4$, $(C_3^2\times S_3^3):C_{12}$, $C_3^5:(D_4\times D_6)$

Order 23040: $C_2\wr A_6$ x 2, $C_2^5.S_6$ x 2, $C_2^5:S_6$ x 2

Order 22464: $C_2\times {\rm PSL}(3,3).C_2$

Order 20736: $C_2\times S_3\wr D_4$

Order 20160: $A_8$

Order 19683: $C_3^4.C_3^4.C_3$

Order 19440: $C_3^4:(C_2\times S_5)$ x 8, $C_3^4:(C_2^2\times A_5)$ x 2, $C_2\times C_3^4:S_5$ x 2, $C_2\times C_3^4:S_5$ x 2, $C_3^4:(D_6\times F_5)$, $C_3:S_3^4:C_5$

Order 18954: $C_3^3.C_3^3.C_{26}$

Order 17496: $C_3^3.C_3^4.C_2^3$ x 14, $C_3^4.C_3^3.C_2^3$ x 12, $C_3^4.S_3^3$ x 10, $C_3^3.C_3^4.D_4$ x 10, $He_3.C_3^4.D_4$ x 7, $He_3.C_3^4.C_2^2.C_2$ x 7, $C_3.C_3^5.D_6.C_2$ x 7, $(C_3\times {\rm He}_3).S_3^3$ x 7, $C_3.C_3^5.D_{12}$ x 6, $C_3.C_3^5.C_6.C_2^2$ x 6, $He_3.C_3^4.Q_8$ x 4, $He_3.C_3^4.C_8$ x 4, $C_3^3:C_3^2.C_6^2.C_2$ x 4, $He_3:(C_3:S_3).C_6^2$ x 3, $C_3.C_3^5.C_{12}.C_2$ x 3, $C_3^5.C_3^2.C_4.C_2$ x 2, $C_3^5.C_3^2.C_2^3$ x 2, $C_3^3.C_3^3.C_{12}.C_2$ x 2, $(C_3^2\times {\rm He}_3).Q_8.C_3^2$ x 2, $He_3.C_3^4.C_4.C_2$, $He_3.C_3^4.C_2^3$, $C_3^4.Q_8.C_3^3$, $C_3^4.C_3^3.C_4.C_2$, $C_3^4.C_3.D_6.C_2$, $C_3^3.C_3^4.Q_8$, $C_3^3.C_3^4.C_4.C_2$, $C_3^3.C_3^4.C_2^2.C_2$, $C_3^3.C_3^3.D_6.C_2$, $C_3^3:\He_3:S_4$, $C_3^3:\He_3:S_4$, $C_3^3.S_3\wr C_3$

Order 17280: $S_4\times S_6$ x 4, $C_2\times A_4:S_6$, $C_2\times S_4\times A_6$, $C_2\times A_4\times S_6$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 9170703360: $\SO(7,3)$ ($C_1$)

Order 4585351680: $B(3,3)$ ($C_2$)

Order 1: $C_1$ ($\SO(7,3)$)

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 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 $98 \times 98$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $95 \times 95$ rational character table.