Abstract group 1451520.a: $\Sp(6,2)$

• Label: 1451520.a
• Order: \( 2^{9} \cdot 3^{4} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Simple: yes
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $28$
• Trans deg.: $28$
• Rank: $2$

Group information

Description:	$\Sp(6,2)$
Order:	\(1451520\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5 \cdot 7 \)
Exponent:	\(2520\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Automorphism group:	$\Sp(6,2)$, of order \(1451520\)\(\medspace = 2^{9} \cdot 3^{4} \cdot 5 \cdot 7 \)
Composition factors:	$\Sp(6,2)$
Derived length:	$0$

This group is nonabelian, simple (hence nonsolvable, perfect, quasisimple, and almost simple), and rational.

Group statistics

Order	1	2	3	4	5	6	7	8	9	10	12	15
Elements	1	5103	16352	75600	48384	272160	207360	181440	161280	145152	241920	96768	1451520
Conjugacy classes	1	4	3	5	1	7	1	2	1	1	3	1	30
Divisions	1	4	3	5	1	7	1	2	1	1	3	1	30
Autjugacy classes	1	4	3	5	1	7	1	2	1	1	3	1	30

Dimension	1	7	15	21	27	35	56	70	84	105	120	168	189	210	216	280	315	336	378	405	420	512
Irr. complex chars.	1	1	1	2	1	2	1	1	1	3	1	1	3	2	1	2	1	1	1	1	1	1	30
Irr. rational chars.	1	1	1	2	1	2	1	1	1	3	1	1	3	2	1	2	1	1	1	1	1	1	30

Minimal presentations

Permutation degree:	$28$
Transitive degree:	$28$
Rank:	$2$
Inequivalent generating pairs:	$1318218$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	7	7	7
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\Sp(6,2)$, $\SO(7,2)$, $\PSp(6,2)$, $\GO(7,2)$, $\Omega(7,2)$, $\PGO(7,2)$, $\Spin(7,2)$, $\CSp(6,2)$, $\PSigmaSp(6,2)$
Permutation group:	Degree $28$ $\langle(1,6,19)(2,18,25,3,12,22)(4,21,7,5,20,13)(8,17,26,15,10,24)(9,11,23,14,16,27), (1,2)(3,15)(4,16)(5,17)(6,18)(9,28)(10,27)(11,24)(12,21)(13,14)\rangle$
Transitive group:	28T1256	36T40610			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as matrices in $\Sp(6,2)$.

Homology

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

Subgroups

There are 11094174 subgroups in 1369 conjugacy classes, 2 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\Sp(6,2)$
Commutator:	$G' \simeq$ $\Sp(6,2)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $\Sp(6,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_1$	$G/\operatorname{Fit} \simeq$ $\Sp(6,2)$
Radical:	$R \simeq$ $C_1$	$G/R \simeq$ $\Sp(6,2)$
Socle:	$\operatorname{soc} \simeq$ $\Sp(6,2)$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

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

Order 51840: $\SO(5,3)$

Order 40320: $S_8$

Order 25920: $\SU(4,2)$

Order 23040: $C_2^5:S_6$

Order 20160: $A_8$

Order 12096: $G(2,2)$

Order 11520: $C_2^5:A_6$

Order 10752: $C_2^6:\GL(3,2)$

Order 6048: $\SU(3,3)$

Order 5040: $S_7$

Order 4608: $C_2^3.\POPlus(4,3)$

Order 4320: $S_3\times S_6$

Order 3840: $C_2^5:S_5$, $C_2\wr S_5$

Order 2520: $A_7$

Order 2304: $C_2^3.A_4\wr C_2$, $C_2^3.A_4\wr C_2$, $C_2^2:\SOPlus(4,3)$, $S_4^2:C_2^2$

Order 2160: $C_3:S_6$, $S_3\times A_6$, $C_3\times S_6$

Order 1920: $C_2^4:S_5$ x 2, $C_2^5:A_5$, $C_2\wr A_5$

Order 1536: $C_2^6:S_4$, $C_2\wr S_4$, $C_2^5:(C_2\times S_4)$

Order 1512: ${}^2G(2,3)$

Order 1440: $C_2\times S_6$ x 2

Order 1344: $C_2^3.\GL(3,2)$, $C_2^3:F_8:C_3$, $C_2^3:\GL(3,2)$

Order 1296: $S_3\wr S_3$, $\He_3:\GL(2,3)$

Order 1152: $S_4\wr C_2$ x 4, $A_4^2:C_2^3$, $C_2\times S_4^2$, $C_2\times A_4^2:C_4$, $Q_8.A_4^2$, $\GOrthPlus(4,3)$

Order 1080: $C_3\times A_6$

Order 960: $C_2^4:A_5$

Order 768: $C_2^5:S_4$ x 2, $C_2^5:S_4$, $C_2^5.S_4$, $C_2^5:S_4$, $Q_8:\GL(2,\mathbb{Z}/4)$, $Q_8.\GL(2,\mathbb{Z}/4)$, $C_2^5:S_4$, $(C_2^3\times C_4):S_4$, $C_2^5:S_4$, $(C_2^3\times C_4):S_4$, $C_2^6:A_4$, $C_2\wr A_4$, $(D_4\times C_2^3):A_4$

Order 720: $S_6$ x 4, $S_3\times S_5$ x 2, $C_2\times A_6$ x 2

Order 648: $S_3\wr C_3$, $C_3^3:S_4$, $C_3^3:S_4$, $\Unitary(3,2)$

Order 640: $C_2\wr F_5$

Order 576: $\POPlus(4,3)$ x 3, $S_4^2$ x 3, $A_4^2:C_4$ x 2, $\OmegaPlus(4,3):C_2$ x 2, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $\SOPlus(4,3)$, $A_4:\GL(2,3)$

Order 512: $C_2^6:D_4$

Order 504: $\SL(2,8)$

Order 448: $C_2^3:F_8$

Order 432: $S_3^3:C_2$, $\He_3:\SD_{16}$

Order 384: $C_2\wr S_4$ x 4, $C_2^4:S_4$ x 2, $(C_2^3\times C_4):A_4$, $C_2^5:A_4$, $C_2^5:A_4$, $C_2^2\wr S_3$, $C_2^5:A_4$, $C_2^5:A_4$, $Q_8.(C_2^2\times A_4)$, $C_2^4.S_4$, $Q_8.(C_2\times S_4)$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^4.S_4$, $C_2^2:\Unitary(2,3)$, $C_2^4.S_4$, $C_2^4:S_4$

Order 360: $S_3\times A_5$ x 2, $C_3:S_5$ x 2, $C_3\times S_5$ x 2, $A_6$ x 2

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

Order 324: $C_3^3:D_6$, $C_3^3:A_4$

Order 320: $C_2^4:F_5$ x 2, $C_2\wr D_5$

Order 288: $D_6\times S_4$ x 2, $\PSOPlus(4,3)$ x 2, $A_4\wr C_2$ x 2, $A_4\times S_4$ x 2, $\OmegaPlus(4,3)$, $A_4\times \SL(2,3)$, $C_2\times A_4^2$

Order 256: $C_2^5.D_4$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$ x 2, $D_4^2:C_2^2$ x 2, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^6:C_4$, $C_2^5:D_4$, $C_2^2:D_4^2$, $C_2^5:D_4$, $C_2^5.D_4$

Order 240: $C_2\times S_5$ x 3

Order 216: $S_3^3$ x 2, $S_3^2:S_3$ x 2, $\SU(3,2)$, $\He_3:D_4$, $\He_3:C_8$, $C_3^2:D_{12}$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$

Order 192: $C_2^3:S_4$ x 8, $D_4\times S_4$ x 8, $C_4^2:D_6$ x 4, $C_2^3:S_4$ x 4, $C_2\wr A_4$ x 4, $Q_8:S_4$ x 4, $C_2^3:S_4$ x 4, $\SL(2,3):D_4$ x 2, $C_4^2:D_6$ x 2, $C_2^3\times S_4$ x 2, $Q_8:S_4$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^2:\GL(2,3)$, $Q_8:S_4$, $C_2^3.\SL(2,3)$, $C_2^2\wr C_3$, $C_2^4:A_4$, $Q_8.(C_2\times A_4)$, $C_2^5:C_6$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_2^4.A_4$, $C_2^4:A_4$

Order 180: $\GL(2,4)$ x 2

Order 168: $F_8:C_3$ x 2, $\PSL(2,7)$ x 2

Order 162: $C_3^3:S_3$, $C_3^3:C_6$, $C_3\wr S_3$

Order 160: $C_2^4:D_5$ x 2, $C_2\wr C_5$

Order 144: $S_3\times S_4$ x 8, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2, $S_3^2:C_2^2$, $A_4^2$, $C_3:\GL(2,3)$

Order 128: $C_2\wr D_4$ x 8, $C_2^4:D_4$ x 8, $C_2^5:C_4$ x 6, $C_2^4.D_4$ x 3, $C_4^2:C_2^3$ x 3, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3.C_4^2$ x 2, $C_2^4.D_4$, $C_2^5.C_4$, $C_2^5:C_4$, $D_4:C_2^4$, $C_2\times D_4^2$, $C_2^4:D_4$, $C_2^4.D_4$, $C_2^5:C_4$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2^4:D_4$, $C_2^4:D_4$, $C_2.D_4^2$

Order 120: $S_5$ x 6, $C_2\times A_5$ x 4, $S_3\times F_5$

Order 108: $C_3\times S_3^2$ x 3, $C_3:S_3^2$ x 2, $C_3:S_3^2$ x 2, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:D_6$, $\He_3:C_4$

Order 96: $C_2^2\times S_4$ x 18, $C_2^2:S_4$ x 8, $\GL(2,\mathbb{Z}/4)$ x 8, $Q_8:A_4$ x 5, $C_4:S_4$ x 5, $C_4^2:S_3$ x 4, $C_2^3:A_4$ x 4, $D_4\times A_4$ x 4, $C_4\times S_4$ x 4, $C_4^2:C_6$ x 2, $C_2^3:A_4$ x 2, $\Unitary(2,3)$ x 2, $C_2^3\times A_4$ x 2, $Q_8:A_4$ x 2, $\SL(2,3):C_2^2$ x 2, $\GL(2,3):C_2$ x 2, $C_4^2:C_6$, $\SL(2,3):C_4$, $A_4:C_8$, $C_2^3.A_4$, $D_4\times D_6$, $Q_8\times A_4$, $C_2^2\times \SL(2,3)$, $C_2^3:C_{12}$, $C_2^2.S_4$, $C_2\times \GL(2,3)$

Order 81: $C_3\wr C_3$

Order 80: $C_2^4:C_5$

Order 72: $S_3\times A_4$ x 4, $C_3:S_4$ x 4, $C_3\times S_4$ x 4, $\SOPlus(4,2)$ x 4, $S_3\times D_6$ x 3, $C_6\times A_4$ x 2, $C_2\times C_3^2:C_4$, $C_3\times \SL(2,3)$

Order 64: $C_2\wr C_2^2$ x 21, $C_2^3:D_4$ x 15, $C_2^4:C_4$ x 10, $C_2\wr C_4$ x 10, $D_4:D_4$ x 10, $D_4^2$ x 6, $C_2^3.D_4$ x 4, $C_4^2:C_4$ x 4, $D_4:C_2^3$ x 4, $C_4^2:C_2^2$ x 4, $\OD_{16}:C_2^2$ x 3, $C_2^3:D_4$ x 3, $D_4\times C_2^3$ x 3, $C_2^3:D_4$ x 3, $C_2^3.D_4$ x 2, $C_2^3.D_4$ x 2, $C_2^3.C_2^3$ x 2, $C_8:C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_2^4:C_4$ x 2, $D_4:D_4$ x 2, $C_2^2:\OD_{16}$, $C_2^3.Q_8$, $C_2^3.C_2^3$, $C_2^3.D_4$, $C_2^4:C_4$, $C_2^2:C_4^2$, $C_2^6$, $D_4:C_2^3$, $C_4^2:C_2^2$, $Q_8:D_4$, $D_4:D_4$

Order 60: $A_5$ x 4, $S_3\times D_5$, $C_{15}:C_4$, $C_3\times F_5$

Order 56: $F_8$ x 2

Order 54: $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2:S_3$, $C_9:C_6$, $C_3^2:C_6$, $C_3^2:S_3$

Order 48: $C_2\times S_4$ x 41, $C_2^2\times A_4$ x 11, $S_3\times D_4$ x 8, $\GL(2,3)$ x 5, $\SL(2,3):C_2$ x 3, $C_4\times A_4$ x 3, $C_2^2\times D_6$ x 2, $C_2^2:A_4$ x 2, $C_6:D_4$ x 2, $A_4:C_4$ x 2, $C_6\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_2\times \SL(2,3)$, $C_4^2:C_3$, $Q_8:S_3$

Order 42: $F_7$

Order 40: $C_2\times F_5$

Order 36: $S_3^2$ x 11, $C_6\times S_3$ x 4, $C_3^2:C_4$ x 2, $C_6:S_3$ x 2, $C_3\times A_4$ x 2

Order 32: $C_2^2\wr C_2$ x 34, $C_2^2\times D_4$ x 32, $C_2^3:C_4$ x 19, $C_2^3:C_4$ x 16, $D_4:C_2^2$ x 16, $C_4:D_4$ x 10, $D_8:C_2$ x 8, $C_4\wr C_2$ x 8, $\OD_{16}:C_2$ x 6, $C_2^5$ x 6, $D_4:C_2^2$ x 6, $C_4:D_4$ x 6, $C_4\times D_4$ x 4, $C_2^3\times C_4$ x 3, $C_2.C_4^2$ x 3, $D_4:C_4$ x 2, $C_2^2:C_8$ x 2, $C_2\times \SD_{16}$ x 2, $C_2\times D_8$ x 2, $C_2\times \OD_{16}$ x 2, $C_2^2.D_4$ x 2, $C_2\times C_4^2$ x 2, $Q_8:C_4$ x 2, $C_2^2\times Q_8$

Order 30: $D_{15}$, $C_3\times D_5$, $C_5\times S_3$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $S_4$ x 21, $C_2\times A_4$ x 19, $C_2\times D_6$ x 12, $C_3:D_4$ x 8, $D_{12}$ x 5, $\SL(2,3)$ x 5, $C_4\times S_3$ x 4, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $C_3\times Q_8$, $C_3:C_8$

Order 21: $C_7:C_3$

Order 20: $F_5$ x 2, $D_{10}$

Order 18: $C_3\times S_3$ x 10, $C_3:S_3$ x 5, $C_3\times C_6$ x 2, $D_9$

Order 16: $C_2\times D_4$ x 79, $C_2^2:C_4$ x 25, $C_2^4$ x 24, $D_4:C_2$ x 16, $C_2^2\times C_4$ x 16, $\SD_{16}$ x 4, $D_8$ x 4, $\OD_{16}$ x 4, $C_4:C_4$ x 3, $C_4^2$ x 3, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2

Order 15: $C_{15}$

Order 14: $D_7$

Order 12: $D_6$ x 19, $A_4$ x 7, $C_2\times C_6$ x 6, $C_{12}$ x 3, $C_3:C_4$ x 2

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 3, $C_9$

Order 8: $C_2^3$ x 40, $D_4$ x 32, $C_2\times C_4$ x 20, $Q_8$ x 4, $C_8$ x 2

Order 7: $C_7$

Order 6: $S_3$ x 9, $C_6$ x 7

Order 5: $C_5$

Order 4: $C_2^2$ x 19, $C_4$ x 5

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 51840: $\SO(5,3)$

Order 40320: $S_8$

Order 25920: $\SU(4,2)$

Order 23040: $C_2^5:S_6$

Order 20160: $A_8$

Order 12096: $G(2,2)$

Order 11520: $C_2^5:A_6$

Order 10752: $C_2^6:\GL(3,2)$

Order 6048: $\SU(3,3)$

Order 5040: $S_7$

Order 4608: $C_2^3.\POPlus(4,3)$

Order 4320: $S_3\times S_6$

Order 3840: $C_2^5:S_5$, $C_2\wr S_5$

Order 2520: $A_7$

Order 2304: $C_2^3.A_4\wr C_2$, $C_2^3.A_4\wr C_2$, $C_2^2:\SOPlus(4,3)$, $S_4^2:C_2^2$

Order 2160: $C_3:S_6$, $S_3\times A_6$, $C_3\times S_6$

Order 1920: $C_2^4:S_5$ x 2, $C_2^5:A_5$, $C_2\wr A_5$

Order 1536: $C_2^6:S_4$, $C_2\wr S_4$, $C_2^5:(C_2\times S_4)$

Order 1512: ${}^2G(2,3)$

Order 1440: $C_2\times S_6$ x 2

Order 1344: $C_2^3.\GL(3,2)$, $C_2^3:F_8:C_3$, $C_2^3:\GL(3,2)$

Order 1296: $S_3\wr S_3$, $\He_3:\GL(2,3)$

Order 1152: $S_4\wr C_2$ x 4, $A_4^2:C_2^3$, $C_2\times S_4^2$, $C_2\times A_4^2:C_4$, $Q_8.A_4^2$, $\GOrthPlus(4,3)$

Order 1080: $C_3\times A_6$

Order 960: $C_2^4:A_5$

Order 768: $C_2^5:S_4$ x 2, $C_2^5:S_4$, $C_2^5.S_4$, $C_2^5:S_4$, $Q_8:\GL(2,\mathbb{Z}/4)$, $Q_8.\GL(2,\mathbb{Z}/4)$, $C_2^5:S_4$, $(C_2^3\times C_4):S_4$, $C_2^5:S_4$, $(C_2^3\times C_4):S_4$, $C_2^6:A_4$, $C_2\wr A_4$, $(D_4\times C_2^3):A_4$

Order 720: $S_6$ x 4, $S_3\times S_5$ x 2, $C_2\times A_6$ x 2

Order 648: $S_3\wr C_3$, $C_3^3:S_4$, $C_3^3:S_4$, $\Unitary(3,2)$

Order 640: $C_2\wr F_5$

Order 576: $\POPlus(4,3)$ x 3, $S_4^2$ x 3, $A_4^2:C_4$ x 2, $\OmegaPlus(4,3):C_2$ x 2, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $A_4^2:C_2^2$, $\SOPlus(4,3)$, $A_4:\GL(2,3)$

Order 512: $C_2^6:D_4$

Order 504: $\SL(2,8)$

Order 448: $C_2^3:F_8$

Order 432: $S_3^3:C_2$, $\He_3:\SD_{16}$

Order 384: $C_2\wr S_4$ x 4, $C_2^4:S_4$ x 2, $(C_2^3\times C_4):A_4$, $C_2^5:A_4$, $C_2^5:A_4$, $C_2^2\wr S_3$, $C_2^5:A_4$, $C_2^5:A_4$, $Q_8.(C_2^2\times A_4)$, $C_2^4.S_4$, $Q_8.(C_2\times S_4)$, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_2^4.S_4$, $C_2^2:\Unitary(2,3)$, $C_2^4.S_4$, $C_2^4:S_4$

Order 360: $S_3\times A_5$ x 2, $C_3:S_5$ x 2, $C_3\times S_5$ x 2, $A_6$ x 2

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

Order 324: $C_3^3:D_6$, $C_3^3:A_4$

Order 320: $C_2^4:F_5$ x 2, $C_2\wr D_5$

Order 288: $D_6\times S_4$ x 2, $\PSOPlus(4,3)$ x 2, $A_4\wr C_2$ x 2, $A_4\times S_4$ x 2, $\OmegaPlus(4,3)$, $A_4\times \SL(2,3)$, $C_2\times A_4^2$

Order 256: $C_2^5.D_4$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$ x 2, $D_4^2:C_2^2$ x 2, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^6:C_4$, $C_2^5:D_4$, $C_2^2:D_4^2$, $C_2^5:D_4$, $C_2^5.D_4$

Order 240: $C_2\times S_5$ x 3

Order 216: $S_3^3$ x 2, $S_3^2:S_3$ x 2, $\SU(3,2)$, $\He_3:D_4$, $\He_3:C_8$, $C_3^2:D_{12}$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$

Order 192: $C_2^3:S_4$ x 8, $D_4\times S_4$ x 8, $C_4^2:D_6$ x 4, $C_2^3:S_4$ x 4, $C_2\wr A_4$ x 4, $Q_8:S_4$ x 4, $C_2^3:S_4$ x 4, $\SL(2,3):D_4$ x 2, $C_4^2:D_6$ x 2, $C_2^3\times S_4$ x 2, $Q_8:S_4$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^2:\GL(2,3)$, $Q_8:S_4$, $C_2^3.\SL(2,3)$, $C_2^2\wr C_3$, $C_2^4:A_4$, $Q_8.(C_2\times A_4)$, $C_2^5:C_6$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_2^4.A_4$, $C_2^4:A_4$

Order 180: $\GL(2,4)$ x 2

Order 168: $F_8:C_3$ x 2, $\PSL(2,7)$ x 2

Order 162: $C_3^3:S_3$, $C_3^3:C_6$, $C_3\wr S_3$

Order 160: $C_2^4:D_5$ x 2, $C_2\wr C_5$

Order 144: $S_3\times S_4$ x 8, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2, $S_3^2:C_2^2$, $A_4^2$, $C_3:\GL(2,3)$

Order 128: $C_2\wr D_4$ x 8, $C_2^4:D_4$ x 8, $C_2^5:C_4$ x 6, $C_2^4.D_4$ x 3, $C_4^2:C_2^3$ x 3, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3.C_4^2$ x 2, $C_2^4.D_4$, $C_2^5.C_4$, $C_2^5:C_4$, $D_4:C_2^4$, $C_2\times D_4^2$, $C_2^4:D_4$, $C_2^4.D_4$, $C_2^5:C_4$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2^4:D_4$, $C_2^4:D_4$, $C_2.D_4^2$

Order 120: $S_5$ x 6, $C_2\times A_5$ x 4, $S_3\times F_5$

Order 108: $C_3\times S_3^2$ x 3, $C_3:S_3^2$ x 2, $C_3:S_3^2$ x 2, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:D_6$, $\He_3:C_4$

Order 96: $C_2^2\times S_4$ x 18, $C_2^2:S_4$ x 8, $\GL(2,\mathbb{Z}/4)$ x 8, $Q_8:A_4$ x 5, $C_4:S_4$ x 5, $C_4^2:S_3$ x 4, $C_2^3:A_4$ x 4, $D_4\times A_4$ x 4, $C_4\times S_4$ x 4, $C_4^2:C_6$ x 2, $C_2^3:A_4$ x 2, $\Unitary(2,3)$ x 2, $C_2^3\times A_4$ x 2, $Q_8:A_4$ x 2, $\SL(2,3):C_2^2$ x 2, $\GL(2,3):C_2$ x 2, $C_4^2:C_6$, $\SL(2,3):C_4$, $A_4:C_8$, $C_2^3.A_4$, $D_4\times D_6$, $Q_8\times A_4$, $C_2^2\times \SL(2,3)$, $C_2^3:C_{12}$, $C_2^2.S_4$, $C_2\times \GL(2,3)$

Order 81: $C_3\wr C_3$

Order 80: $C_2^4:C_5$

Order 72: $S_3\times A_4$ x 4, $C_3:S_4$ x 4, $C_3\times S_4$ x 4, $\SOPlus(4,2)$ x 4, $S_3\times D_6$ x 3, $C_6\times A_4$ x 2, $C_2\times C_3^2:C_4$, $C_3\times \SL(2,3)$

Order 64: $C_2\wr C_2^2$ x 21, $C_2^3:D_4$ x 15, $C_2^4:C_4$ x 10, $C_2\wr C_4$ x 10, $D_4:D_4$ x 10, $D_4^2$ x 6, $C_2^3.D_4$ x 4, $C_4^2:C_4$ x 4, $D_4:C_2^3$ x 4, $C_4^2:C_2^2$ x 4, $\OD_{16}:C_2^2$ x 3, $C_2^3:D_4$ x 3, $D_4\times C_2^3$ x 3, $C_2^3:D_4$ x 3, $C_2^3.D_4$ x 2, $C_2^3.D_4$ x 2, $C_2^3.C_2^3$ x 2, $C_8:C_2^3$ x 2, $C_4^2:C_2^2$ x 2, $C_2^4:C_4$ x 2, $D_4:D_4$ x 2, $C_2^2:\OD_{16}$, $C_2^3.Q_8$, $C_2^3.C_2^3$, $C_2^3.D_4$, $C_2^4:C_4$, $C_2^2:C_4^2$, $C_2^6$, $D_4:C_2^3$, $C_4^2:C_2^2$, $Q_8:D_4$, $D_4:D_4$

Order 60: $A_5$ x 4, $S_3\times D_5$, $C_{15}:C_4$, $C_3\times F_5$

Order 56: $F_8$ x 2

Order 54: $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_3^2:S_3$, $C_9:C_6$, $C_3^2:C_6$, $C_3^2:S_3$

Order 48: $C_2\times S_4$ x 41, $C_2^2\times A_4$ x 11, $S_3\times D_4$ x 8, $\GL(2,3)$ x 5, $\SL(2,3):C_2$ x 3, $C_4\times A_4$ x 3, $C_2^2\times D_6$ x 2, $C_2^2:A_4$ x 2, $C_6:D_4$ x 2, $A_4:C_4$ x 2, $C_6\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_2\times \SL(2,3)$, $C_4^2:C_3$, $Q_8:S_3$

Order 42: $F_7$

Order 40: $C_2\times F_5$

Order 36: $S_3^2$ x 11, $C_6\times S_3$ x 4, $C_3^2:C_4$ x 2, $C_6:S_3$ x 2, $C_3\times A_4$ x 2

Order 32: $C_2^2\wr C_2$ x 34, $C_2^2\times D_4$ x 32, $C_2^3:C_4$ x 19, $C_2^3:C_4$ x 16, $D_4:C_2^2$ x 16, $C_4:D_4$ x 10, $D_8:C_2$ x 8, $C_4\wr C_2$ x 8, $\OD_{16}:C_2$ x 6, $C_2^5$ x 6, $D_4:C_2^2$ x 6, $C_4:D_4$ x 6, $C_4\times D_4$ x 4, $C_2^3\times C_4$ x 3, $C_2.C_4^2$ x 3, $D_4:C_4$ x 2, $C_2^2:C_8$ x 2, $C_2\times \SD_{16}$ x 2, $C_2\times D_8$ x 2, $C_2\times \OD_{16}$ x 2, $C_2^2.D_4$ x 2, $C_2\times C_4^2$ x 2, $Q_8:C_4$ x 2, $C_2^2\times Q_8$

Order 30: $D_{15}$, $C_3\times D_5$, $C_5\times S_3$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $S_4$ x 21, $C_2\times A_4$ x 19, $C_2\times D_6$ x 12, $C_3:D_4$ x 8, $D_{12}$ x 5, $\SL(2,3)$ x 5, $C_4\times S_3$ x 4, $C_3\times D_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $C_3\times Q_8$, $C_3:C_8$

Order 21: $C_7:C_3$

Order 20: $F_5$ x 2, $D_{10}$

Order 18: $C_3\times S_3$ x 10, $C_3:S_3$ x 5, $C_3\times C_6$ x 2, $D_9$

Order 16: $C_2\times D_4$ x 79, $C_2^2:C_4$ x 25, $C_2^4$ x 24, $D_4:C_2$ x 16, $C_2^2\times C_4$ x 16, $\SD_{16}$ x 4, $D_8$ x 4, $\OD_{16}$ x 4, $C_4:C_4$ x 3, $C_4^2$ x 3, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2

Order 15: $C_{15}$

Order 14: $D_7$

Order 12: $D_6$ x 19, $A_4$ x 7, $C_2\times C_6$ x 6, $C_{12}$ x 3, $C_3:C_4$ x 2

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 3, $C_9$

Order 8: $C_2^3$ x 40, $D_4$ x 32, $C_2\times C_4$ x 20, $Q_8$ x 4, $C_8$ x 2

Order 7: $C_7$

Order 6: $S_3$ x 9, $C_6$ x 7

Order 5: $C_5$

Order 4: $C_2^2$ x 19, $C_4$ x 5

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1451520: $\Sp(6,2)$ ($C_1$)

Order 1: $C_1$ ($\Sp(6,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1451520: $\Sp(6,2)$ ($C_1$)

Order 1: $C_1$ ($\Sp(6,2)$)

Series

Derived series	$\Sp(6,2)$
Chief series	$\Sp(6,2)$	$\rhd$	$C_1$
Lower central series	$\Sp(6,2)$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $30 \times 30$ rational character table. Alternatively, you may search for characters of this group with desired properties.