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G = C8xM4(2)  order 128 = 27

Direct product of C8 and M4(2)

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C8xM4(2), C82:14C2, C23.24C42, C42.741C23, C8:6(C2xC8), C8o3(C4:C8), C4.9(C4xC8), C8o(C8:C8), C4:C8.25C4, (C4xC8).16C4, C8o2(C8:C4), C8:C8:24C2, C8o2(C22:C8), C22.9(C4xC8), C8:C4.20C4, C4.51(C8oD4), C22:C8.23C4, (C22xC8).26C4, C4.32(C22xC8), (C2xC4).58C42, C2.1(C4xM4(2)), (C4xC8).441C22, C42.241(C2xC4), C4.56(C2xM4(2)), C8o(C42.12C4), (C2xM4(2)).36C4, (C4xM4(2)).32C2, C22.21(C2xC42), C2.1(C8o2M4(2)), C42.12C4.49C2, (C2xC42).1028C22, C2.2(C2xC4xC8), (C2xC4xC8).7C2, (C4xC8)o(C8:C4), (C2xC8)o(C8:C8), (C4xC8)o(C8:C8), (C2xC4).49(C2xC8), (C2xC8)o(C4xM4(2)), (C4xC8)o(C2xM4(2)), (C2xC8).262(C2xC4), (C2xC4).578(C22xC4), (C22xC4).430(C2xC4), (C2xC8)o(C42.12C4), SmallGroup(128,181)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C8xM4(2)
C1C2C22C2xC4C42C2xC42C2xC4xC8 — C8xM4(2)
C1C2 — C8xM4(2)
C1C4xC8 — C8xM4(2)
C1C22C22C42 — C8xM4(2)

Generators and relations for C8xM4(2)
 G = < a,b,c | a8=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 132 in 112 conjugacy classes, 92 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, C23, C42, C2xC8, C2xC8, M4(2), C22xC4, C4xC8, C4xC8, C8:C4, C22:C8, C4:C8, C2xC42, C22xC8, C2xM4(2), C82, C8:C8, C2xC4xC8, C4xM4(2), C42.12C4, C8xM4(2)
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, C42, C2xC8, M4(2), C22xC4, C4xC8, C2xC42, C22xC8, C2xM4(2), C8oD4, C2xC4xC8, C4xM4(2), C8o2M4(2), C8xM4(2)

Smallest permutation representation of C8xM4(2)
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 48 10 61 31 23 35 52)(2 41 11 62 32 24 36 53)(3 42 12 63 25 17 37 54)(4 43 13 64 26 18 38 55)(5 44 14 57 27 19 39 56)(6 45 15 58 28 20 40 49)(7 46 16 59 29 21 33 50)(8 47 9 60 30 22 34 51)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R8A···8P8Q···8BD
order1222224···44···48···88···8
size1111221···12···21···12···2

80 irreducible representations

dim111111111111122
type++++++
imageC1C2C2C2C2C2C4C4C4C4C4C4C8M4(2)C8oD4
kernelC8xM4(2)C82C8:C8C2xC4xC8C4xM4(2)C42.12C4C4xC8C8:C4C22:C8C4:C8C22xC8C2xM4(2)M4(2)C8C4
# reps1221114444443288

Matrix representation of C8xM4(2) in GL3(F17) generated by

1500
090
009
,
1300
0013
0160
,
100
0160
001
G:=sub<GL(3,GF(17))| [15,0,0,0,9,0,0,0,9],[13,0,0,0,0,16,0,13,0],[1,0,0,0,16,0,0,0,1] >;

C8xM4(2) in GAP, Magma, Sage, TeX

C_8\times M_4(2)
% in TeX

G:=Group("C8xM4(2)");
// GroupNames label

G:=SmallGroup(128,181);
// by ID

G=gap.SmallGroup(128,181);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,120,758,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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