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G = C8×M4(2)  order 128 = 27

Direct product of C8 and M4(2)

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

Aliases: C8×M4(2), C8214C2, C23.24C42, C42.741C23, C86(C2×C8), C83(C4⋊C8), C4.9(C4×C8), C8(C8⋊C8), C4⋊C8.25C4, (C4×C8).16C4, C82(C8⋊C4), C8⋊C824C2, C82(C22⋊C8), C22.9(C4×C8), C8⋊C4.20C4, C4.51(C8○D4), C22⋊C8.23C4, (C22×C8).26C4, C4.32(C22×C8), (C2×C4).58C42, C2.1(C4×M4(2)), (C4×C8).441C22, C42.241(C2×C4), C4.56(C2×M4(2)), C8(C42.12C4), (C2×M4(2)).36C4, (C4×M4(2)).32C2, C22.21(C2×C42), C2.1(C82M4(2)), C42.12C4.49C2, (C2×C42).1028C22, C2.2(C2×C4×C8), (C2×C4×C8).7C2, (C4×C8)(C8⋊C4), (C2×C8)(C8⋊C8), (C4×C8)(C8⋊C8), (C2×C4).49(C2×C8), (C2×C8)(C4×M4(2)), (C4×C8)(C2×M4(2)), (C2×C8).262(C2×C4), (C2×C4).578(C22×C4), (C22×C4).430(C2×C4), (C2×C8)(C42.12C4), SmallGroup(128,181)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C8×M4(2)
C1C2C22C2×C4C42C2×C42C2×C4×C8 — C8×M4(2)
C1C2 — C8×M4(2)
C1C4×C8 — C8×M4(2)
C1C22C22C42 — C8×M4(2)

Generators and relations for C8×M4(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 [×3], C2 [×2], C4 [×8], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×12], C8 [×6], C2×C4 [×6], C2×C4 [×4], C2×C4 [×4], C23, C42 [×4], C2×C8 [×12], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], C4×C8 [×2], C4×C8 [×6], C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C22×C8 [×2], C2×M4(2) [×2], C82 [×2], C8⋊C8 [×2], C2×C4×C8, C4×M4(2), C42.12C4, C8×M4(2)
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], M4(2) [×4], C22×C4 [×3], C4×C8 [×4], C2×C42, C22×C8 [×2], C2×M4(2) [×2], C8○D4 [×2], C2×C4×C8, C4×M4(2), C82M4(2), C8×M4(2)

Smallest permutation representation of C8×M4(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)C8○D4
kernelC8×M4(2)C82C8⋊C8C2×C4×C8C4×M4(2)C42.12C4C4×C8C8⋊C4C22⋊C8C4⋊C8C22×C8C2×M4(2)M4(2)C8C4
# reps1221114444443288

Matrix representation of C8×M4(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] >;

C8×M4(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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