Abstract group 56899584.f: $C_2\times \PSL(2,7)\wr S_3$

• Label: 56899584.f
• Order: \( 2^{11} \cdot 3^{4} \cdot 7^{3} \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{4} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $23$
• Trans deg.: $42$
• Rank: $2$

Group information

Description:	$C_2\times \PSL(2,7)\wr S_3$
Order:	\(56899584\)\(\medspace = 2^{11} \cdot 3^{4} \cdot 7^{3} \)
Exponent:	\(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \)
Automorphism group:	$C_2\times \PSL(2,7)^3.D_6$, of order \(113799168\)\(\medspace = 2^{12} \cdot 3^{4} \cdot 7^{3} \)
Composition factors:	$C_2$ x 2, $C_3$, $\PSL(2,7)$ x 3
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	8	9	12	14	18	21	24	28	42	56	84
Elements	1	43471	241640	1900080	7629272	117648	2709504	3161088	10301760	3982320	3161088	3564288	2370816	4669056	9660672	2032128	1354752	56899584
Conjugacy classes	1	11	4	20	20	9	6	1	14	43	1	9	2	26	21	4	4	196
Divisions	1	11	4	20	20	5	6	1	14	23	1	5	2	14	11	2	2	142
Autjugacy classes	1	9	4	16	16	5	3	1	12	18	1	5	1	12	9	1	2	116

Dimension	1	2	9	18	21	24	27	54	81	108	126	144	147	162	189	192	216	252	288	324	336	343	378	432	441	512	576	648	686	756	864	882	1008	1024	1152	1176	1344	1512	1728	2016
Irr. complex chars.	4	2	8	4	4	4	16	6	8	8	4	4	4	8	8	4	12	2	2	10	2	4	2	4	8	4	8	0	2	8	8	4	4	2	4	4	4	0	0	2	196
Irr. rational chars.	4	2	0	8	4	4	0	10	0	6	0	0	4	4	0	4	6	4	4	6	2	4	6	8	0	4	0	4	2	4	4	8	0	2	8	4	4	2	2	4	142

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$42$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	18
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $23$ $\langle(1,3)(2,5,8,13,18,21,11,17)(4,6,10,16)(7,12,19,20)(9,15)(22,23), (1,2,4)(3,6,9,14,8,10)(5,7,11)(12,18,21,19,15,13)(16,17,20)(22,23)\rangle$
Transitive group:	42T3420	42T3421			more information
Direct product:	$C_2$ $\, \times\, $ $(\PSL(2,7)\wr S_3)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$\PSL(2,7)^3$ . $D_6$	$(\PSL(2,7)\wr S_3)$ . $C_2$ (2)	$C_2$ . $(\PSL(2,7)\wr S_3)$	$(\PSL(2,7)\wr C_3)$ . $C_2^2$	all 6

Elements of the group are displayed as permutations of degree 23.

Homology

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

Subgroups

There are 9 normal subgroups (7 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 $\PSL(2,7)\wr C_3$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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 6144 (order at least 9261) are shown, as well as all normal subgroups of any index.

Order 56899584: $C_2\times \PSL(2,7)\wr S_3$

Order 28449792: $\PSL(2,7)\wr S_3$ x 2, $C_2\times \PSL(2,7)\wr C_3$

Order 18966528: $(C_2\times \PSL(2,7)).\PSL(2,7)^2.C_2$

Order 14224896: $\PSL(2,7)\wr C_3$

Order 9483264: $\PSL(2,7).\PSL(2,7)^2.C_2$ x 2, $(C_2\times \PSL(2,7)).\PSL(2,7).\PSL(2,7)$

Order 4741632: $\PSL(2,7)^3$

Order 2709504: $(C_2\times S_4).\PSL(2,7)^2.C_2$ x 2

Order 2370816: $(C_2\times C_7:C_3).\PSL(2,7)^2.C_2$

Order 1354752: $S_4.\PSL(2,7)^2.C_2$ x 8, $(C_2\times A_4).\PSL(2,7)^2.C_2$ x 4, $(C_2\times S_4).\PSL(2,7).\PSL(2,7)$ x 2

Order 1185408: $C_7:C_3.\PSL(2,7)^2.C_2$ x 2, $(C_2\times C_7:C_3).\PSL(2,7).\PSL(2,7)$

Order 903168: $(C_2\times D_4).\PSL(2,7)^2.C_2$

Order 790272: $C_{14}.\PSL(2,7)^2.C_2$

Order 677376: $A_4.\PSL(2,7)^2.C_2$ x 8, $\PSL(2,7)^2\times S_4$ x 2, $S_4\times \PSL(2,7)^2$ x 2, $(C_2\times A_4).\PSL(2,7).\PSL(2,7)$ x 2, $C_2\times S_3.\PSL(2,7)^2.C_2$

Order 592704: $C_7:C_3\times \PSL(2,7)^2$

Order 451584: $D_4.\PSL(2,7)^2.C_2$ x 8, $C_2^3.\PSL(2,7)^2.C_2$ x 4, $(C_2\times C_4).\PSL(2,7)^2.C_2$ x 2, $(C_2\times D_4).\PSL(2,7).\PSL(2,7)$

Order 395136: $C_7.\PSL(2,7)^2.C_2$ x 2, $C_{14}.\PSL(2,7).\PSL(2,7)$

Order 387072: $A_4^2.C_2^3.C_2.\PSL(2,7)$ x 2

Order 338688: $\GL(3,2)^2:D_6$ x 4, $C_6.\PSL(2,7)^2.C_2$ x 2, $A_4.\PSL(2,7).\PSL(2,7)$ x 2, $D_6.\PSL(2,7).\PSL(2,7)$

Order 296352: $C_7^2.(C_6\times S_3).\PSL(2,7)$

Order 225792: $C_4.\PSL(2,7)^2.C_2$ x 8, $C_2^2\times \PSL(2,7)^2.C_2$ x 8, $\GL(3,2)^2:D_4$ x 8, $D_4.\PSL(2,7).\PSL(2,7)$ x 4, $C_2^2.\PSL(2,7)^2.C_2$ x 4, $C_2^3.\PSL(2,7).\PSL(2,7)$ x 2, $(C_2\times C_4).\PSL(2,7).\PSL(2,7)$

Order 197568: $C_7.\PSL(2,7).\PSL(2,7)$

Order 193536: $S_4\wr C_2\times \GL(3,2)$ x 8, $A_4^2.C_2^3.\PSL(2,7)$ x 5, $A_4^2.C_2^2.C_2.\PSL(2,7)$ x 2

Order 169344: $S_3.\PSL(2,7).\PSL(2,7)$ x 2, $(A_4\times C_7:C_3).C_2^2.\PSL(2,7)$ x 2, $\GL(3,2)^2:S_3$ x 2, $\GL(3,2)^2:C_6$ x 2, $C_6.\PSL(2,7).\PSL(2,7)$

Order 165888: $S_4^3.D_6$ x 2

Order 148176: $C_7^2.(C_3\times S_3).\PSL(2,7)$ x 2, $C_2\times C_7^2:C_3:C_3.\PSL(2,7)$

Order 112896: $\GL(3,2)^2:C_2^2$ x 12, $C_2^2.\PSL(2,7).\PSL(2,7)$ x 5, $C_4.\PSL(2,7).\PSL(2,7)$ x 2, $\GL(3,2)^2:C_4$ x 2, $C_2\times (C_2\times \PSL(2,7)).\PSL(2,7)$, $(C_2\times \PSL(2,7)).\PSL(2,7).C_2$

Order 111132: $C_7^3:(C_3^3:D_6)$

Order 98784: $C_7^2:C_3.C_2^2.\PSL(2,7)$, $C_7^2.D_6.\PSL(2,7)$

Order 96768: $S_4^2\times \GL(3,2)$ x 10, $A_4^2.C_2^2.\PSL(2,7)$ x 9, $\GL(3,2)\times \POPlus(4,3)$ x 6, $A_4^2:C_4\times \GL(3,2)$ x 4

Order 84672: $(C_7:C_3\times S_4).\PSL(2,7)$ x 4, $(C_2\times A_4\times C_7:C_3).\PSL(2,7)$ x 2, $C_3\times \PSL(2,7)^2$

Order 82944: $A_4^3.(C_2\times S_4)$ x 8, $C_2\times A_4^3:S_4$ x 2, $C_2\times A_4^3:S_4$ x 2, $S_4^3.C_6$ x 2

Order 74088: $C_7^2:C_3:C_3.\PSL(2,7)$

Order 64512: $C_2^4:C_3.C_2^3.\PSL(2,7)$ x 2, $(C_2^3\times S_4).C_2.\PSL(2,7)$ x 2

Order 56448: $(C_2\times \PSL(2,7)).\PSL(2,7)$ x 7, $\GL(3,2)\wr C_2$ x 4, $(C_{14}\times S_4).\PSL(2,7)$ x 2, $\GL(3,2)\times \PGL(2,7)$ x 2, $(C_2\times D_4\times C_7:C_3).\PSL(2,7)$

Order 55566: $C_7^3:C_3\wr S_3$ x 2, $C_7^3:\He_3:C_6$

Order 55296: $S_4^3.C_2^2$ x 4

Order 49392: $C_7^2:S_3.\PSL(2,7)$ x 2, $C_7^2:C_6.\PSL(2,7)$ x 2, $(C_{14}\times C_7:C_3).\PSL(2,7)$, $(C_2\times C_7^2:C_3).\PSL(2,7)$, $(C_2\times C_7:(C_7:C_3)).\PSL(2,7)$

Order 48384: $A_4\times S_4\times \GL(3,2)$ x 8, $\GL(3,2)\times \PSOPlus(4,3)$ x 6, $A_4\wr C_2\times \GL(3,2)$ x 4, $(C_2\times A_4^2).\PSL(2,7)$ x 3, $A_4^2.C_7.C_6.C_2^3$ x 2, $(C_2\times S_3\times S_4).\PSL(2,7)$ x 2

Order 43008: $C_2^3.C_2^4.C_2.\PSL(2,7)$

Order 42336: $(A_4\times C_7:C_3).\PSL(2,7)$ x 2, $C_{14}^2.C_3^3.C_2^3$, $C_7^2.A_4.C_6^2.C_2$, $(C_2\times S_3\times C_7:C_3).\PSL(2,7)$

Order 41472: $S_4\wr C_3$ x 4, $A_4^3:S_4$ x 4, $A_4^3:S_4$ x 4, $C_2\times A_4^3:A_4$ x 2, $C_2\times A_4^3:D_6$ x 2

Order 37044: $C_7^3.C_3.C_6^2$, $C_7^3:C_3^2:D_6$

Order 32928: $(C_{14}\times D_7).\PSL(2,7)$

Order 32256: $D_4\times S_4\times \GL(3,2)$ x 16, $C_2^3:S_4\times \GL(3,2)$ x 8, $(C_2\times C_2^2:S_4).\PSL(2,7)$ x 5, $(C_2^3\times S_4).\PSL(2,7)$ x 4, $(C_2\times \GL(2,\mathbb{Z}/4)).\PSL(2,7)$ x 4, $(C_2\times D_4\times A_4).\PSL(2,7)$ x 2, $(C_2\times C_4\times S_4).\PSL(2,7)$ x 2, $(C_2\times C_4:S_4).\PSL(2,7)$ x 2, $(C_2\times C_2^2\wr C_2:C_3).\PSL(2,7)$ x 2

Order 28224: $(D_4\times C_7:C_3).\PSL(2,7)$ x 4, $(C_7\times S_4).\PSL(2,7)$ x 4, $\GL(3,2)^2$ x 3, $(C_{14}\times A_4).\PSL(2,7)$ x 2, $(C_2^3\times C_7:C_3).\PSL(2,7)$ x 2, $(C_2\times C_7:A_4).\PSL(2,7)$ x 2, $(C_2\times C_4\times C_7:C_3).\PSL(2,7)$

Order 27783: $C_7^3:C_3\wr C_3$

Order 27648: $S_4^3:C_2$ x 32, $C_2\times S_4^3$ x 4, $C_2\times A_4^2:\GL(2,\mathbb{Z}/4)$ x 4, $C_2\times A_4^3:D_4$ x 4, $C_2\times S_4^2:S_4$ x 4, $A_4^3.C_2^4$ x 4, $A_4^3.(C_2^2\times C_4)$ x 4, $C_2\times A_4^3:D_4$ x 4

Order 24696: $C_7^2.(C_3\times \PSL(2,7))$ x 2, $C_7^2:C_3.\PSL(2,7)$

Order 24192: $S_3\times S_4\times \GL(3,2)$ x 8, $A_4^2:(C_{28}:C_6)$ x 8, $(C_2\times C_{14}\times A_4).C_6^2.C_2$ x 3, $A_4^2\times \GL(3,2)$ x 3, $C_2^4.(C_3\times C_{21}).C_6.C_2^2$ x 2, $(C_6\times S_4).\PSL(2,7)$ x 2, $(C_2^3\times C_{14}).C_3^3.C_2^3$ x 2, $(C_2\times S_3\times A_4).\PSL(2,7)$ x 2, $(C_2\times C_3:S_4).\PSL(2,7)$ x 2, $S_3^2:C_2^2\times \PSL(2,7)$

Order 21504: $C_2\wr D_4\times \PSL(2,7)$ x 8, $C_2^5.C_2^2.\PSL(2,7)$ x 3, $(C_2\times D_4).C_2^3.\PSL(2,7)$ x 3, $(C_2\times C_4).C_2^4.\PSL(2,7)$

Order 21168: $C_{14}^2.C_3^3.C_2^2$ x 10, $C_7^2.(C_6\times S_3\times A_4)$ x 2, $(S_3\times C_7:C_3).\PSL(2,7)$ x 2, $C_{14}^2.C_3.C_6^2$, $C_2\times C_{14}^2.C_3^3.C_2$, $(C_6\times C_7:C_3).\PSL(2,7)$

Order 20736: $A_4^3:D_6$ x 8, $A_4^3:D_6$ x 2, $C_2\times A_4^3:C_6$ x 2, $A_4^3:A_4$ x 2, $A_4^3:D_6$ x 2

Order 18816: $(C_{14}\times D_4).\PSL(2,7)$

Order 18522: $C_7^3:(S_3\times C_3^2)$ x 2, $C_7^3:C_3^2:S_3$ x 2, $C_7^3:(C_3^2\times C_6)$, $C_7^3:C_9:C_6$, $C_2\times C_7^3:\He_3$

Order 18432: $C_2\times A_4^2.D_4^2$ x 2

Order 16464: $(C_7\times D_7).\PSL(2,7)$ x 2, $(C_7\times C_{14}).\PSL(2,7)$

Order 16128: $C_2^2\times S_4\times \GL(3,2)$ x 26, $\GL(2,\mathbb{Z}/4)\times \GL(3,2)$ x 16, $C_2^2:S_4\times \GL(3,2)$ x 10, $C_4:S_4\times \PSL(2,7)$ x 8, $C_4\times S_4\times \GL(3,2)$ x 8, $D_4\times A_4\times \GL(3,2)$ x 8, $(C_2^3\times A_4).\PSL(2,7)$ x 4, $C_2^3:A_4\times \GL(3,2)$ x 4, $(C_2\times C_2^4:C_3).\PSL(2,7)$ x 3, $A_4^2.C_{14}.C_2^3$ x 2, $(C_2\times C_4\times A_4).\PSL(2,7)$ x 2, $(C_2\times A_4:C_4).\PSL(2,7)$ x 2, $C_2\times S_4\times \PGL(2,7)$ x 2, $(C_2\times S_3\times D_4).\PSL(2,7)$

Order 14112: $(C_2^2\times C_7:C_3).\PSL(2,7)$ x 6, $C_{14}^2.C_3^2.C_2^3$ x 2, $C_7^2.(C_6\times S_4).C_2$ x 2, $C_7:A_4.\PSL(2,7)$ x 2, $(C_7\times A_4).\PSL(2,7)$ x 2, $(C_4\times C_7:C_3).\PSL(2,7)$ x 2, $C_7^2.C_6^2.C_2^3$, $C_2\times C_7:C_3.\SO(3,7)$, $C_2\times (C_2\times C_7:C_3).\PSL(2,7)$, $(C_{14}\times S_3).\PSL(2,7)$, $(C_2\times F_7).\PSL(2,7)$

Order 13824: $A_4^3.C_2^3$ x 24, $S_4^3$ x 20, $A_4^2:C_4\times S_4$ x 16, $A_4^2:\GL(2,\mathbb{Z}/4)$ x 16, $A_4^3:D_4$ x 16, $S_4^2:S_4$ x 16, $A_4^3:D_4$ x 16, $A_4^3.C_2^3$ x 6, $A_4^3.C_2^3$ x 6, $A_4^3.C_2^3$ x 4, $A_4^3.C_2^3$ x 4, $C_2\times A_4^3:C_4$ x 4, $A_4^3:C_2^3$ x 4, $A_4^3.C_2^3$ x 4, $C_2\times A_4^3:C_4$ x 4, $A_4^3:C_2^3$ x 4, $C_2^3:A_4^2:D_6$ x 2, $D_6\times S_4\wr C_2$ x 2

Order 12348: $C_7^3:(C_6\times S_3)$ x 3, $C_7^3:(C_6\times S_3)$ x 2, $C_7^3:C_6^2$

Order 12096: $C_7:C_3\times S_4^2$ x 10, $A_4^2:C_{14}:C_6$ x 6, $(C_7\times A_4).A_4.C_6.C_2$ x 4, $(C_7\times A_4^2):C_{12}$ x 4, $\GL(3,2)\times \SOPlus(4,2)$ x 4, $C_3:S_4\times \GL(3,2)$ x 4, $C_3\times S_4\times \GL(3,2)$ x 4, $S_3\times A_4\times \GL(3,2)$ x 4, $C_2^4.(C_3\times C_{21}).C_6.C_2$ x 3, $(C_7\times C_2^4:C_3).C_6^2$ x 2, $(C_6\times A_4).\PSL(2,7)$ x 2, $S_3\times D_6\times \GL(3,2)$ x 2, $C_2\times C_3^2:C_4\times \GL(3,2)$

Order 10752: $C_2\wr C_2^2\times \GL(3,2)$ x 12, $D_4^2.\PSL(2,7)$ x 10, $C_2\wr C_4\times \PSL(2,7)$ x 8, $D_4:D_4\times \PSL(2,7)$ x 6, $(C_2\times C_2^2.D_4).\PSL(2,7)$ x 4, $C_4^2:C_4\times \GL(3,2)$ x 4, $C_2^5.(C_2\times \PSL(2,7))$ x 3, $(C_2^3\times D_4).\PSL(2,7)$ x 2, $(C_2\times C_2^2\wr C_2).\PSL(2,7)$ x 2, $(C_2\times C_2^2:D_4).\PSL(2,7)$ x 2, $(C_2\times Q_8:C_2^2).\PSL(2,7)$, $(C_2\times C_8:C_2^2).\PSL(2,7)$, $(C_2\times C_4\wr C_2).\PSL(2,7)$, $(C_2\times C_4\times D_4).\PSL(2,7)$, $(C_2\times C_4:D_4).\PSL(2,7)$, $(C_2\times C_2^3.C_4).\PSL(2,7)$

Order 10584: $C_{14}^2.C_3^3.C_2$ x 10, $C_7^2.(C_3\times A_4).C_6$ x 4, $C_7^2.C_3^3.C_2^3$, $(C_3\times C_7:C_3).\PSL(2,7)$

Order 10368: $A_4^3:S_3$ x 4, $A_4\wr S_3$ x 4, $A_4^3:C_6$ x 4, $A_4^3:C_6$ x 2

Order 9408: $(C_7\times D_4).\PSL(2,7)$ x 4, $(C_2^2\times C_{14}).\PSL(2,7)$ x 2, $(C_2\times C_{28}).\PSL(2,7)$

Order 9261: $C_7^3:C_3^3$, $C_7^3:C_9:C_3$, $C_7^3:\He_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 56899584: $C_2\times \PSL(2,7)\wr S_3$ ($C_1$)

Order 28449792: $\PSL(2,7)\wr S_3$ ($C_2$) x 2, $C_2\times \PSL(2,7)\wr C_3$ ($C_2$)

Order 14224896: $\PSL(2,7)\wr C_3$ ($C_2^2$)

Order 9483264: $(C_2\times \PSL(2,7)).\PSL(2,7).\PSL(2,7)$ ($S_3$)

Order 4741632: $\PSL(2,7)^3$ ($D_6$)

Order 2: $C_2$ ($\PSL(2,7)\wr S_3$)

Order 1: $C_1$ ($C_2\times \PSL(2,7)\wr S_3$)

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 3 larger groups in the database.

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

Character theory

Complex character table

See the $196 \times 196$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $142 \times 142$ rational character table (warning: may be slow to load).