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G = C4.A4:C4order 192 = 26·3

4th semidirect product of C4.A4 and C4 acting via C4/C2=C2

non-abelian, soluble

Aliases: C4.A4:4C4, (C2xC4).26S4, C4.8(A4:C4), Q8:Dic3:9C2, C4oD4:1Dic3, (C2xQ8).16D6, C22.19(C2xS4), Q8.4(C2xDic3), C2.3(C4.6S4), SL2(F3).7(C2xC4), (C2xSL2(F3)).16C22, C2.9(C2xA4:C4), (C2xC4.A4).6C2, (C2xC4oD4).3S3, SmallGroup(192,983)

Series: Derived Chief Lower central Upper central

C1C2Q8SL2(F3) — C4.A4:C4
C1C2Q8SL2(F3)C2xSL2(F3)Q8:Dic3 — C4.A4:C4
SL2(F3) — C4.A4:C4
C1C2xC4

Generators and relations for C4.A4:C4
 G = < a,b,c,d,e | a4=d3=e4=1, b2=c2=a2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ece-1=a2b, dbd-1=a2bc, ebe-1=a2c, dcd-1=b, ede-1=d-1 >

Subgroups: 267 in 79 conjugacy classes, 21 normal (11 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C8, C2xC4, C2xC4, D4, Q8, Q8, C23, Dic3, C12, C2xC6, C42, C22:C4, C4:C4, C2xC8, C22xC4, C2xD4, C2xQ8, C4oD4, C4oD4, SL2(F3), C2xDic3, C2xC12, D4:C4, Q8:C4, C42:C2, C22xC8, C2xC4oD4, C4xDic3, C2xSL2(F3), C4.A4, C23.24D4, Q8:Dic3, C2xC4.A4, C4.A4:C4
Quotients: C1, C2, C4, C22, S3, C2xC4, Dic3, D6, C2xDic3, S4, A4:C4, C2xS4, C4.6S4, C2xA4:C4, C4.A4:C4

Smallest permutation representation of C4.A4:C4
On 64 points
Generators in S64
(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 31 3 29)(2 32 4 30)(5 16 7 14)(6 13 8 15)(9 43 11 41)(10 44 12 42)(17 23 19 21)(18 24 20 22)(25 61 27 63)(26 62 28 64)(33 54 35 56)(34 55 36 53)(37 59 39 57)(38 60 40 58)(45 50 47 52)(46 51 48 49)
(1 9 3 11)(2 10 4 12)(5 21 7 23)(6 22 8 24)(13 20 15 18)(14 17 16 19)(25 52 27 50)(26 49 28 51)(29 43 31 41)(30 44 32 42)(33 38 35 40)(34 39 36 37)(45 61 47 63)(46 62 48 64)(53 57 55 59)(54 58 56 60)
(5 21 14)(6 22 15)(7 23 16)(8 24 13)(9 43 31)(10 44 32)(11 41 29)(12 42 30)(25 52 63)(26 49 64)(27 50 61)(28 51 62)(33 60 40)(34 57 37)(35 58 38)(36 59 39)
(1 53 17 45)(2 54 18 46)(3 55 19 47)(4 56 20 48)(5 27 41 37)(6 28 42 38)(7 25 43 39)(8 26 44 40)(9 36 16 52)(10 33 13 49)(11 34 14 50)(12 35 15 51)(21 61 29 57)(22 62 30 58)(23 63 31 59)(24 64 32 60)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I4J6A6B6C8A···8H12A12B12C12D
order122222344444444446668···812121212
size1111668111166121212128886···68888

32 irreducible representations

dim111122223334
type+++++-++
imageC1C2C2C4S3D6Dic3C4.6S4S4A4:C4C2xS4C4.6S4
kernelC4.A4:C4Q8:Dic3C2xC4.A4C4.A4C2xC4oD4C2xQ8C4oD4C2C2xC4C4C22C2
# reps121411282424

Matrix representation of C4.A4:C4 in GL4(F73) generated by

72000
07200
00460
00046
,
1000
0100
006658
00527
,
1000
0100
002265
001551
,
72100
72000
00721
00720
,
02700
27000
002842
007045
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,46,0,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,66,52,0,0,58,7],[1,0,0,0,0,1,0,0,0,0,22,15,0,0,65,51],[72,72,0,0,1,0,0,0,0,0,72,72,0,0,1,0],[0,27,0,0,27,0,0,0,0,0,28,70,0,0,42,45] >;

C4.A4:C4 in GAP, Magma, Sage, TeX

C_4.A_4\rtimes C_4
% in TeX

G:=Group("C4.A4:C4");
// GroupNames label

G:=SmallGroup(192,983);
// by ID

G=gap.SmallGroup(192,983);
# by ID

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

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

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