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G = (C2×C4).S4order 192 = 26·3

15th non-split extension by C2×C4 of S4 acting via S4/A4=C2

non-abelian, soluble

Aliases: C4.A42C4, (C2×C4).15S4, C4.3(A4⋊C4), Q8⋊Dic37C2, C4○D42Dic3, (C2×Q8).18D6, C22.20(C2×S4), Q8.5(C2×Dic3), C2.2(C4.3S4), C2.2(C4.S4), SL2(𝔽3)⋊5(C2×C4), (C2×SL2(𝔽3)).18C22, C2.10(C2×A4⋊C4), (C2×C4.A4).2C2, (C2×C4○D4).5S3, SmallGroup(192,985)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(𝔽3) — (C2×C4).S4
C1C2Q8SL2(𝔽3)C2×SL2(𝔽3)Q8⋊Dic3 — (C2×C4).S4
SL2(𝔽3) — (C2×C4).S4
C1C22C2×C4

Generators and relations for (C2×C4).S4
 G = < a,b,c,d,e,f | a2=b4=e3=1, c2=d2=b2, f2=a, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=b-1, dcd-1=b2c, ece-1=b2cd, fcf-1=cd, ede-1=c, fdf-1=b2d, fef-1=e-1 >

Subgroups: 291 in 79 conjugacy classes, 21 normal (13 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×4], C6 [×3], C8 [×2], C2×C4, C2×C4 [×9], D4 [×3], Q8, Q8, C23, Dic3 [×2], C12 [×2], C2×C6, C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C22×C4 [×2], C2×D4, C2×Q8, C4○D4 [×2], C4○D4 [×2], SL2(𝔽3), C2×Dic3 [×2], C2×C12, D4⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C2×C4○D4, C4⋊Dic3, C2×SL2(𝔽3), C4.A4 [×2], C23.36D4, Q8⋊Dic3 [×2], C2×C4.A4, (C2×C4).S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, Dic3 [×2], D6, C2×Dic3, S4, A4⋊C4 [×2], C2×S4, C4.S4, C4.3S4, C2×A4⋊C4, (C2×C4).S4

Character table of (C2×C4).S4

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H6A6B6C8A8B8C8D12A12B12C12D
 size 1111668226612121212888121212128888
ρ111111111111111111111111111    trivial
ρ21111-1-11-1-1111-11-111111-1-1-1-1-1-1    linear of order 2
ρ311111111111-1-1-1-1111-1-1-1-11111    linear of order 2
ρ41111-1-11-1-111-11-11111-1-111-1-1-1-1    linear of order 2
ρ51-1-111-11-111-1i-i-ii-11-1i-i-ii-1-111    linear of order 4
ρ61-1-11-1111-11-1ii-i-i-11-1i-ii-i11-1-1    linear of order 4
ρ71-1-111-11-111-1-iii-i-11-1-iii-i-1-111    linear of order 4
ρ81-1-11-1111-11-1-i-iii-11-1-ii-ii11-1-1    linear of order 4
ρ9222222-122220000-1-1-10000-1-1-1-1    orthogonal lifted from S3
ρ102222-2-2-1-2-2220000-1-1-100001111    orthogonal lifted from D6
ρ112-2-22-22-12-22-200001-110000-1-111    symplectic lifted from Dic3, Schur index 2
ρ122-2-222-2-1-222-200001-11000011-1-1    symplectic lifted from Dic3, Schur index 2
ρ133333-1-1033-1-11111000-1-1-1-10000    orthogonal lifted from S4
ρ143333110-3-3-1-1-11-1100011-1-10000    orthogonal lifted from C2×S4
ρ153333110-3-3-1-11-11-1000-1-1110000    orthogonal lifted from C2×S4
ρ163333-1-1033-1-1-1-1-1-100011110000    orthogonal lifted from S4
ρ173-3-331-103-3-11ii-i-i000-ii-ii0000    complex lifted from A4⋊C4
ρ183-3-33-110-33-11-iii-i000i-i-ii0000    complex lifted from A4⋊C4
ρ193-3-33-110-33-11i-i-ii000-iii-i0000    complex lifted from A4⋊C4
ρ203-3-331-103-3-11-i-iii000i-ii-i0000    complex lifted from A4⋊C4
ρ214-44-400-20000000022-200000000    orthogonal lifted from C4.3S4
ρ224-44-400100000000-1-110000-33-33    orthogonal lifted from C4.3S4
ρ234-44-400100000000-1-1100003-33-3    orthogonal lifted from C4.3S4
ρ2444-4-400-200000000-22200000000    symplectic lifted from C4.S4, Schur index 2
ρ2544-4-4001000000001-1-100003-3-33    symplectic lifted from C4.S4, Schur index 2
ρ2644-4-4001000000001-1-10000-333-3    symplectic lifted from C4.S4, Schur index 2

Smallest permutation representation of (C2×C4).S4
On 64 points
Generators in S64
(1 7)(2 8)(3 5)(4 6)(9 21)(10 22)(11 23)(12 24)(13 36)(14 33)(15 34)(16 35)(17 46)(18 47)(19 48)(20 45)(25 29)(26 30)(27 31)(28 32)(37 41)(38 42)(39 43)(40 44)(49 56)(50 53)(51 54)(52 55)(57 61)(58 62)(59 63)(60 64)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 9 3 11)(2 10 4 12)(5 23 7 21)(6 24 8 22)(13 25 15 27)(14 26 16 28)(17 52 19 50)(18 49 20 51)(29 34 31 36)(30 35 32 33)(37 58 39 60)(38 59 40 57)(41 62 43 64)(42 63 44 61)(45 54 47 56)(46 55 48 53)
(1 15 3 13)(2 16 4 14)(5 36 7 34)(6 33 8 35)(9 25 11 27)(10 26 12 28)(17 44 19 42)(18 41 20 43)(21 29 23 31)(22 30 24 32)(37 45 39 47)(38 46 40 48)(49 64 51 62)(50 61 52 63)(53 57 55 59)(54 58 56 60)
(9 15 27)(10 16 28)(11 13 25)(12 14 26)(17 52 63)(18 49 64)(19 50 61)(20 51 62)(21 34 31)(22 35 32)(23 36 29)(24 33 30)(45 54 58)(46 55 59)(47 56 60)(48 53 57)
(1 42 7 38)(2 41 8 37)(3 44 5 40)(4 43 6 39)(9 50 21 53)(10 49 22 56)(11 52 23 55)(12 51 24 54)(13 17 36 46)(14 20 33 45)(15 19 34 48)(16 18 35 47)(25 63 29 59)(26 62 30 58)(27 61 31 57)(28 64 32 60)

G:=sub<Sym(64)| (1,7)(2,8)(3,5)(4,6)(9,21)(10,22)(11,23)(12,24)(13,36)(14,33)(15,34)(16,35)(17,46)(18,47)(19,48)(20,45)(25,29)(26,30)(27,31)(28,32)(37,41)(38,42)(39,43)(40,44)(49,56)(50,53)(51,54)(52,55)(57,61)(58,62)(59,63)(60,64), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,9,3,11)(2,10,4,12)(5,23,7,21)(6,24,8,22)(13,25,15,27)(14,26,16,28)(17,52,19,50)(18,49,20,51)(29,34,31,36)(30,35,32,33)(37,58,39,60)(38,59,40,57)(41,62,43,64)(42,63,44,61)(45,54,47,56)(46,55,48,53), (1,15,3,13)(2,16,4,14)(5,36,7,34)(6,33,8,35)(9,25,11,27)(10,26,12,28)(17,44,19,42)(18,41,20,43)(21,29,23,31)(22,30,24,32)(37,45,39,47)(38,46,40,48)(49,64,51,62)(50,61,52,63)(53,57,55,59)(54,58,56,60), (9,15,27)(10,16,28)(11,13,25)(12,14,26)(17,52,63)(18,49,64)(19,50,61)(20,51,62)(21,34,31)(22,35,32)(23,36,29)(24,33,30)(45,54,58)(46,55,59)(47,56,60)(48,53,57), (1,42,7,38)(2,41,8,37)(3,44,5,40)(4,43,6,39)(9,50,21,53)(10,49,22,56)(11,52,23,55)(12,51,24,54)(13,17,36,46)(14,20,33,45)(15,19,34,48)(16,18,35,47)(25,63,29,59)(26,62,30,58)(27,61,31,57)(28,64,32,60)>;

G:=Group( (1,7)(2,8)(3,5)(4,6)(9,21)(10,22)(11,23)(12,24)(13,36)(14,33)(15,34)(16,35)(17,46)(18,47)(19,48)(20,45)(25,29)(26,30)(27,31)(28,32)(37,41)(38,42)(39,43)(40,44)(49,56)(50,53)(51,54)(52,55)(57,61)(58,62)(59,63)(60,64), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,9,3,11)(2,10,4,12)(5,23,7,21)(6,24,8,22)(13,25,15,27)(14,26,16,28)(17,52,19,50)(18,49,20,51)(29,34,31,36)(30,35,32,33)(37,58,39,60)(38,59,40,57)(41,62,43,64)(42,63,44,61)(45,54,47,56)(46,55,48,53), (1,15,3,13)(2,16,4,14)(5,36,7,34)(6,33,8,35)(9,25,11,27)(10,26,12,28)(17,44,19,42)(18,41,20,43)(21,29,23,31)(22,30,24,32)(37,45,39,47)(38,46,40,48)(49,64,51,62)(50,61,52,63)(53,57,55,59)(54,58,56,60), (9,15,27)(10,16,28)(11,13,25)(12,14,26)(17,52,63)(18,49,64)(19,50,61)(20,51,62)(21,34,31)(22,35,32)(23,36,29)(24,33,30)(45,54,58)(46,55,59)(47,56,60)(48,53,57), (1,42,7,38)(2,41,8,37)(3,44,5,40)(4,43,6,39)(9,50,21,53)(10,49,22,56)(11,52,23,55)(12,51,24,54)(13,17,36,46)(14,20,33,45)(15,19,34,48)(16,18,35,47)(25,63,29,59)(26,62,30,58)(27,61,31,57)(28,64,32,60) );

G=PermutationGroup([(1,7),(2,8),(3,5),(4,6),(9,21),(10,22),(11,23),(12,24),(13,36),(14,33),(15,34),(16,35),(17,46),(18,47),(19,48),(20,45),(25,29),(26,30),(27,31),(28,32),(37,41),(38,42),(39,43),(40,44),(49,56),(50,53),(51,54),(52,55),(57,61),(58,62),(59,63),(60,64)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,9,3,11),(2,10,4,12),(5,23,7,21),(6,24,8,22),(13,25,15,27),(14,26,16,28),(17,52,19,50),(18,49,20,51),(29,34,31,36),(30,35,32,33),(37,58,39,60),(38,59,40,57),(41,62,43,64),(42,63,44,61),(45,54,47,56),(46,55,48,53)], [(1,15,3,13),(2,16,4,14),(5,36,7,34),(6,33,8,35),(9,25,11,27),(10,26,12,28),(17,44,19,42),(18,41,20,43),(21,29,23,31),(22,30,24,32),(37,45,39,47),(38,46,40,48),(49,64,51,62),(50,61,52,63),(53,57,55,59),(54,58,56,60)], [(9,15,27),(10,16,28),(11,13,25),(12,14,26),(17,52,63),(18,49,64),(19,50,61),(20,51,62),(21,34,31),(22,35,32),(23,36,29),(24,33,30),(45,54,58),(46,55,59),(47,56,60),(48,53,57)], [(1,42,7,38),(2,41,8,37),(3,44,5,40),(4,43,6,39),(9,50,21,53),(10,49,22,56),(11,52,23,55),(12,51,24,54),(13,17,36,46),(14,20,33,45),(15,19,34,48),(16,18,35,47),(25,63,29,59),(26,62,30,58),(27,61,31,57),(28,64,32,60)])

Matrix representation of (C2×C4).S4 in GL7(𝔽73)

72000000
07200000
00720000
00072000
00007200
00000720
00000072
,
72000000
07200000
00720000
000066766
000706666
000667066
0007770
,
07210000
07200000
17200000
0000010
0000001
00072000
00007200
,
01720000
10720000
00720000
00000072
0000010
00007200
0001000
,
0010000
1000000
0100000
0001000
0000001
00007200
00000720
,
02700000
27000000
00270000
0004369469
0006944369
0004436969
00069696930

G:=sub<GL(7,GF(73))| [72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,7,66,7,0,0,0,66,0,7,7,0,0,0,7,66,0,7,0,0,0,66,66,66,0],[0,0,1,0,0,0,0,72,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,72,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0],[0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0],[0,27,0,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,43,69,4,69,0,0,0,69,4,43,69,0,0,0,4,43,69,69,0,0,0,69,69,69,30] >;

(C2×C4).S4 in GAP, Magma, Sage, TeX

(C_2\times C_4).S_4
% in TeX

G:=Group("(C2xC4).S4");
// GroupNames label

G:=SmallGroup(192,985);
// by ID

G=gap.SmallGroup(192,985);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,28,1373,680,451,1684,655,172,1013,404,285,124]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^4=e^3=1,c^2=d^2=b^2,f^2=a,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b^-1,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,f*c*f^-1=c*d,e*d*e^-1=c,f*d*f^-1=b^2*d,f*e*f^-1=e^-1>;
// generators/relations

Export

Character table of (C2×C4).S4 in TeX

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