Abstract group $C_2\times \SO(5,3)$

• Label: 103680.a
• Order: \( 2^{8} \cdot 3^{4} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{8} \cdot 3^{4} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $29$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$C_2\times \SO(5,3)$
Order:	\(103680\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5 \)
Exponent:	\(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$C_2\times \SO(5,3)$, of order \(103680\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5 \)
Outer automorphisms:	$C_2$, of order \(2\)
Composition factors:	$C_2$ x 2, $C(2,3)$
Derived length:	$1$

This group is nonabelian, nonsolvable, and rational.

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	18
Elements	1	1783	800	11880	5184	26720	12960	5760	15552	17280	5760	103680
Conjugacy classes	1	9	3	8	1	17	2	1	3	4	1	50
Divisions	1	9	3	8	1	17	2	1	3	4	1	50
Autjugacy classes	1	7	3	6	1	14	1	1	2	3	1	40

Dimension	1	6	10	15	20	24	30	60	64	80	81	90
Irr. complex chars.	4	4	2	8	6	4	4	6	4	2	4	2	50
Irr. rational chars.	4	4	2	8	6	4	4	6	4	2	4	2	50

Minimal Presentations

Permutation degree:	$29$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$34515$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	6	6
Arbitrary	not computed	not computed	not computed

Constructions

Groups of Lie type:	$\GO(5,3)$
Permutation group:	Degree $29$ $\langle(1,3,8,15)(2,6)(4,10,16,20)(5,11,19,7)(9,12,21,23)(13,22,17,18)(14,24)(25,26) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 & 1 & -1 \\ 0 & 1 & 1 & 1 & 0 & 0 \\ -1 & 0 & -1 & 0 & -1 & 1 \\ 1 & -1 & 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & -1 & -1 & 0 \\ 0 & 1 & 1 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & -1 & -1 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
Direct product:	$C_2$ $\, \times\, $ $\SO(5,3)$
Semidirect product:	$C(2,3)$ $\,\rtimes\,$ $C_2^2$	$(C_2\times C(2,3))$ $\,\rtimes\,$ $C_2$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(\GU(4,2))$	$\Aut(\Sp(4,3):C_2)$

Elements of the group are displayed as matrices in $\GO(5,3)$.

Homology

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

Subgroups

There are 1046205 subgroups in 1503 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:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\SO(5,3)$
Commutator:	$G' \simeq$ $C(2,3)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2\times \SO(5,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_2$	$G/\operatorname{Fit} \simeq$ $\SO(5,3)$
Radical:	$R \simeq$ $C_2$	$G/R \simeq$ $\SO(5,3)$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C(2,3)$	$G/\operatorname{soc} \simeq$ $C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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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 103680: $C_2\times \SO(5,3)$

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

Order 25920: $C(2,3)$

Order 3840: $C_2\wr S_5$

Order 2880: $C_2^2\times S_6$

Order 2592: $S_3^3:D_6$, $\SU(3,2):D_6$

Order 2304: $\SOPlus(4,3):C_2^2$

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

Order 1440: $C_2\times S_6$ x 6, $C_2^2\times A_6$

Order 1296: $S_3\wr S_3$ x 4, $\He_3:\GL(2,3)$ x 2, $S_3^3:C_6$, $C_2\times C_3^3:S_4$, $C_2\times C_3^3:S_4$, $C_2\times \GU(3,2)$

Order 1152: $C_2^3:A_4.D_6$ x 4, $C_2^3:A_4.C_6.C_2$ x 2, $C_2^3:A_4.D_6$

Order 960: $C_2^4:A_5$

Order 864: $S_3^3:C_2^2$, $\SU(3,2):C_2^2$

Order 768: $C_2^5:S_4$ x 2

Order 720: $S_6$ x 4, $C_2\times A_6$ x 3

Order 648: $S_3\wr C_3$ x 2, $C_3^3:S_4$ x 2, $C_3^3:S_4$ x 2, $C_2\times C_3^3:A_4$, $\GU(3,2)$, $C_2\times C_3^3:D_6$

Order 640: $C_2\wr F_5$

Order 576: $\OmegaPlus(4,3):C_2$ x 4, $\SOPlus(4,3)$ x 2, $C_2^2.A_4^2$

Order 480: $C_2^2\times S_5$ x 2

Order 432: $S_3^3:C_2$ x 8, $\He_3:\SD_{16}$ x 4, $C_2\times S_3^3$ x 2, $S_3^2:D_6$ x 2, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$, $C_2\times \SU(3,2)$, $C_2\times \He_3:D_4$, $C_6.F_9$

Order 384: $C_2\wr S_4$ x 8, $C_2^5:A_4$ x 2, $C_2^4.S_4$ x 2, $C_2^4:S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$, $C_4.\GL(2,\mathbb{Z}/4)$

Order 360: $A_6$

Order 324: $C_3^3:D_6$ x 4, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:D_6$, $C_3^3:A_4$

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

Order 288: $C_6:\GL(2,3)$, $\OmegaPlus(4,3)$, $C_6^2:D_4$

Order 256: $D_4^2:C_2^2$

Order 240: $C_2\times S_5$ x 12, $C_2^2\times A_5$ x 2

Order 216: $S_3^3$ x 10, $S_3^2:S_3$ x 8, $C_3^2:D_{12}$ x 4, $S_3^2:C_6$ x 4, $C_3^2:C_4\times S_3$ x 4, $\He_3:D_4$ x 3, $C_6\times S_3^2$ x 3, $\SU(3,2)$ x 2, $\He_3:C_8$ x 2, $C_6:S_3^2$ x 2, $C_6:S_3^2$ x 2, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$, $C_6.S_3^2$, $C_2\times \He_3:C_4$

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

Order 162: $C_3^3:S_3$ x 2, $C_3^3:C_6$ x 2, $C_3\wr S_3$ x 2, $C_3^3:C_6$

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

Order 144: $S_3^2:C_2^2$ x 12, $D_6^2$ x 3, $C_3:\GL(2,3)$ x 2, $C_6^2:C_4$, $C_6\times \SL(2,3)$

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

Order 120: $S_5$ x 8, $C_2\times A_5$ x 6

Order 108: $C_3\times S_3^2$ x 10, $C_3:S_3^2$ x 8, $C_3:S_3^2$ x 5, $C_3^2:D_6$ x 3, $C_3^2:D_6$ x 2, $C_3^2\times D_6$ x 2, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2, $\He_3:C_4$ x 2, $C_3^2:D_6$, $C_3^2:D_6$, $C_{18}:C_6$, $C_3^2:D_6$

Order 96: $C_2^2\times S_4$ x 33, $\GL(2,\mathbb{Z}/4)$ x 8, $D_4\times A_4$ x 4, $C_4:S_4$ x 4, $C_4\times S_4$ x 4, $C_2^3\times A_4$ x 3, $\GL(2,3):C_2$ x 3, $C_2\times \GL(2,3)$ x 3, $\GU(2,3)$ x 2, $Q_8:A_4$ x 2, $\SL(2,3):C_2^2$ x 2, $D_4\times D_6$, $\SL(2,3):C_2^2$, $C_2^3:C_{12}$, $C_2^2.S_4$, $Q_8:D_6$

Order 81: $C_3\wr C_3$

Order 80: $C_2^2\times F_5$, $C_2^4:C_5$

Order 72: $S_3\times D_6$ x 31, $\SOPlus(4,2)$ x 16, $C_2\times C_3^2:C_4$ x 6, $C_6\times D_6$ x 4, $C_6:D_6$ x 2, $C_3\times \SL(2,3)$

Order 64: $D_4^2$ x 10, $C_2\wr C_4$ x 8, $C_2\wr C_2^2$ x 8, $D_4:D_4$ x 6, $C_4^2:C_4$ x 4, $C_2^3:D_4$ x 3, $C_2^4:C_4$ x 2, $D_4\times C_2^3$ x 2, $C_2^3:D_4$ x 2, $\OD_{16}:C_2^2$, $D_4:C_2^3$, $C_8:C_2^3$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$

Order 60: $A_5$ x 2

Order 54: $C_3^2:C_6$ x 4, $S_3\times C_3^2$ x 4, $C_3^2:S_3$ x 2, $C_9:C_6$ x 2, $C_3^2:C_6$ x 2, $C_3^2:S_3$ x 2, $C_3^2\times C_6$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $C_2\times S_4$ x 58, $C_2^2\times A_4$ x 16, $S_3\times D_4$ x 8, $\GL(2,3)$ x 6, $Q_8:S_3$ x 4, $C_2^2\times D_6$ x 3, $C_2\times \SL(2,3)$ x 3, $C_6:D_4$ x 2, $C_2\times D_{12}$ x 2, $\SL(2,3):C_2$ x 2, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_6:C_8$, $C_6\times Q_8$, $C_6\times D_4$, $C_4\times D_6$

Order 40: $C_2\times F_5$ x 6, $C_2\times D_{10}$

Order 36: $S_3^2$ x 40, $C_6\times S_3$ x 26, $C_6:S_3$ x 13, $C_3^2:C_4$ x 4, $C_6^2$ x 2, $D_{18}$

Order 32: $C_2^2\times D_4$ x 33, $C_2^2\wr C_2$ x 20, $C_4:D_4$ x 16, $C_4\times D_4$ x 8, $C_2^3:C_4$ x 6, $D_8:C_2$ x 6, $C_4:D_4$ x 6, $C_2^3:C_4$ x 5, $C_4\wr C_2$ x 4, $\OD_{16}:C_2$ x 3, $C_2^5$ x 3, $D_4:C_2^2$ x 3, $C_2^3\times C_4$ x 2, $D_4:C_2^2$, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_2\times \OD_{16}$, $C_2^2.D_4$, $C_2\times C_4^2$

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

Order 24: $C_2\times D_6$ x 34, $S_4$ x 16, $C_2\times A_4$ x 14, $C_3:D_4$ x 8, $D_{12}$ x 7, $C_4\times S_3$ x 4, $C_3\times D_4$ x 4, $\SL(2,3)$ x 3, $C_2^2\times C_6$ x 3, $C_2\times C_{12}$ x 2, $C_3\times Q_8$ x 2, $C_3:C_8$ x 2, $C_6:C_4$

Order 20: $D_{10}$ x 6, $F_5$ x 4, $C_2\times C_{10}$

Order 18: $C_3\times S_3$ x 20, $C_3:S_3$ x 10, $C_3\times C_6$ x 7, $D_9$ x 2, $C_{18}$

Order 16: $C_2\times D_4$ x 98, $C_2^4$ x 23, $C_2^2:C_4$ x 18, $C_2^2\times C_4$ x 16, $\SD_{16}$ x 4, $D_8$ x 4, $C_4:C_4$ x 4, $D_4:C_2$ x 4, $\OD_{16}$ x 3, $C_4^2$ x 3, $C_2\times C_8$, $C_2\times Q_8$

Order 12: $D_6$ x 62, $C_2\times C_6$ x 17, $C_{12}$ x 4, $A_4$ x 2, $C_3:C_4$ x 2

Order 10: $D_5$ x 4, $C_{10}$ x 3

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

Order 8: $D_4$ x 50, $C_2^3$ x 49, $C_2\times C_4$ x 25, $Q_8$ x 2, $C_8$ x 2

Order 6: $S_3$ x 18, $C_6$ x 17

Order 5: $C_5$

Order 4: $C_2^2$ x 34, $C_4$ x 8

Order 3: $C_3$ x 3

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 103680: $C_2\times \SO(5,3)$

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

Order 25920: $C(2,3)$

Order 3840: $C_2\wr S_5$

Order 2880: $C_2^2\times S_6$

Order 2592: $S_3^3:D_6$, $\SU(3,2):D_6$

Order 2304: $\SOPlus(4,3):C_2^2$

Order 1920: $C_2^5.A_5$, $C_2^4.S_5$

Order 1440: $C_2\times S_6$ x 4, $C_2^2\times A_6$

Order 1296: $S_3\wr S_3$ x 2, $S_3^3:C_6$, $C_2\times C_3^3:S_4$, $C_2\times C_3^3:S_4$, $C_2\times \GU(3,2)$, $\He_3:\GL(2,3)$

Order 1152: $C_2^3:A_4.C_6.C_2$ x 2, $C_2^3:A_4.D_6$ x 2, $C_2^3:A_4.D_6$

Order 960: $C_2^4:A_5$

Order 864: $S_3^3:C_2^2$, $\SU(3,2):C_2^2$

Order 768: $C_2^5:S_4$ x 2

Order 720: $S_6$ x 3, $C_2\times A_6$ x 2

Order 648: $C_3^3:S_4$ x 2, $C_2\times C_3^3:A_4$, $S_3\wr C_3$, $C_3^3:S_4$, $\GU(3,2)$, $C_2\times C_3^3:D_6$

Order 640: $C_2\wr F_5$

Order 576: $\OmegaPlus(4,3):C_2$ x 3, $C_2^2.A_4^2$, $\SOPlus(4,3)$

Order 480: $C_2^2\times S_5$ x 2

Order 432: $S_3^3:C_2$ x 4, $C_2\times S_3^3$ x 2, $S_3^2:D_6$ x 2, $\He_3:\SD_{16}$ x 2, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$, $C_2\times \SU(3,2)$, $C_2\times \He_3:D_4$, $C_6.F_9$

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

Order 360: $A_6$

Order 324: $C_3^3:D_6$ x 2, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:D_6$, $C_3^3:A_4$

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

Order 288: $C_6:\GL(2,3)$, $\OmegaPlus(4,3)$, $C_6^2:D_4$

Order 256: $D_4^2:C_2^2$

Order 240: $C_2\times S_5$ x 8, $C_2^2\times A_5$ x 2

Order 216: $S_3^2:S_3$ x 6, $S_3^3$ x 5, $C_6\times S_3^2$ x 3, $\SU(3,2)$ x 2, $\He_3:D_4$ x 2, $C_6:S_3^2$ x 2, $C_6:S_3^2$ x 2, $C_3^2:D_{12}$ x 2, $S_3^2:C_6$ x 2, $C_3^2:C_4\times S_3$ x 2, $\He_3:C_8$, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$, $C_6.S_3^2$, $C_2\times \He_3:C_4$

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

Order 162: $C_3\wr S_3$ x 2, $C_3^3:C_6$, $C_3^3:S_3$, $C_3^3:C_6$

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

Order 144: $S_3^2:C_2^2$ x 8, $D_6^2$ x 3, $C_6^2:C_4$, $C_6\times \SL(2,3)$, $C_3:\GL(2,3)$

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

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

Order 108: $C_3\times S_3^2$ x 7, $C_3:S_3^2$ x 4, $C_3:S_3^2$ x 4, $C_3^2:D_6$ x 2, $C_3^2\times D_6$ x 2, $C_3^3:C_4$ x 2, $C_3^2:D_6$ x 2, $\He_3:C_4$ x 2, $C_3^2:D_6$, $C_3^2:C_{12}$, $C_3^2:D_6$, $C_{18}:C_6$, $C_3^2:D_6$

Order 96: $C_2^2\times S_4$ x 23, $\GL(2,\mathbb{Z}/4)$ x 6, $C_2^3\times A_4$ x 3, $C_2\times \GL(2,3)$ x 3, $Q_8:A_4$ x 2, $\SL(2,3):C_2^2$ x 2, $D_4\times A_4$ x 2, $\GL(2,3):C_2$ x 2, $C_4:S_4$ x 2, $C_4\times S_4$ x 2, $\GU(2,3)$, $D_4\times D_6$, $\SL(2,3):C_2^2$, $C_2^3:C_{12}$, $C_2^2.S_4$, $Q_8:D_6$

Order 81: $C_3\wr C_3$

Order 80: $C_2^2\times F_5$, $C_2^4:C_5$

Order 72: $S_3\times D_6$ x 21, $\SOPlus(4,2)$ x 10, $C_6\times D_6$ x 4, $C_2\times C_3^2:C_4$ x 4, $C_6:D_6$ x 2, $C_3\times \SL(2,3)$

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

Order 60: $A_5$ x 2

Order 54: $C_3^2:C_6$ x 3, $S_3\times C_3^2$ x 3, $C_3^2:S_3$ x 2, $C_9:C_6$, $C_3^2:C_6$, $C_3^2\times C_6$, $C_3^2:S_3$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $C_2\times S_4$ x 37, $C_2^2\times A_4$ x 13, $S_3\times D_4$ x 4, $C_2^2\times D_6$ x 3, $C_2\times \SL(2,3)$ x 3, $\GL(2,3)$ x 3, $C_6:D_4$ x 2, $C_2\times D_{12}$ x 2, $\SL(2,3):C_2$ x 2, $A_4:C_4$ x 2, $Q_8:S_3$ x 2, $C_6:C_8$, $C_6\times Q_8$, $C_6\times D_4$, $C_4\times D_6$, $C_4\times A_4$

Order 40: $C_2\times F_5$ x 4, $C_2\times D_{10}$

Order 36: $S_3^2$ x 24, $C_6\times S_3$ x 20, $C_6:S_3$ x 9, $C_3^2:C_4$ x 3, $C_6^2$ x 2, $D_{18}$

Order 32: $C_2^2\times D_4$ x 23, $C_2^2\wr C_2$ x 14, $C_4:D_4$ x 8, $C_2^3:C_4$ x 5, $C_2^3:C_4$ x 5, $D_8:C_2$ x 4, $C_4:D_4$ x 4, $C_4\times D_4$ x 4, $C_2^5$ x 3, $D_4:C_2^2$ x 3, $\OD_{16}:C_2$ x 2, $C_2^3\times C_4$ x 2, $C_4\wr C_2$ x 2, $D_4:C_2^2$, $C_2\times \SD_{16}$, $C_2\times D_8$, $C_2\times \OD_{16}$, $C_2^2.D_4$, $C_2\times C_4^2$

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

Order 24: $C_2\times D_6$ x 24, $C_2\times A_4$ x 11, $S_4$ x 11, $C_3:D_4$ x 6, $D_{12}$ x 4, $\SL(2,3)$ x 3, $C_2^2\times C_6$ x 3, $C_2\times C_{12}$ x 2, $C_4\times S_3$ x 2, $C_3\times Q_8$ x 2, $C_3\times D_4$ x 2, $C_6:C_4$, $C_3:C_8$

Order 20: $D_{10}$ x 4, $F_5$ x 3, $C_2\times C_{10}$

Order 18: $C_3\times S_3$ x 15, $C_3\times C_6$ x 6, $C_3:S_3$ x 6, $C_{18}$, $D_9$

Order 16: $C_2\times D_4$ x 63, $C_2^4$ x 19, $C_2^2:C_4$ x 13, $C_2^2\times C_4$ x 11, $D_4:C_2$ x 4, $\SD_{16}$ x 2, $D_8$ x 2, $\OD_{16}$ x 2, $C_4:C_4$ x 2, $C_4^2$ x 2, $C_2\times C_8$, $C_2\times Q_8$

Order 12: $D_6$ x 40, $C_2\times C_6$ x 14, $C_{12}$ x 3, $A_4$ x 2, $C_3:C_4$ x 2

Order 10: $D_5$ x 3, $C_{10}$ x 2

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

Order 8: $C_2^3$ x 37, $D_4$ x 31, $C_2\times C_4$ x 18, $Q_8$ x 2, $C_8$

Order 6: $C_6$ x 14, $S_3$ x 12

Order 5: $C_5$

Order 4: $C_2^2$ x 26, $C_4$ x 6

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 103680: $C_2\times \SO(5,3)$ ($C_1$)

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

Order 25920: $C(2,3)$ ($C_2^2$)

Order 2: $C_2$ ($\SO(5,3)$)

Order 1: $C_1$ ($C_2\times \SO(5,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 103680: $C_2\times \SO(5,3)$ ($C_1$)

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

Order 25920: $C(2,3)$ ($C_2^2$)

Order 2: $C_2$ ($\SO(5,3)$)

Order 1: $C_1$ ($C_2\times \SO(5,3)$)

Series

Derived series	$C_2\times \SO(5,3)$	$\rhd$	$C(2,3)$
Chief series	$C_2\times \SO(5,3)$	$\rhd$	$C_2\times C(2,3)$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$C_2\times \SO(5,3)$	$\rhd$	$C(2,3)$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 0 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 $50 \times 50$ rational character table. Alternatively, you may search for characters of this group with desired properties.