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G = C81M4(2)  order 128 = 27

1st semidirect product of C8 and M4(2) acting via M4(2)/C4=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C81M4(2), C42.89D4, C42.16Q8, C42.626C23, C82C88C2, C81C812C2, C8⋊C4.14C4, (C22×C4).28Q8, C4⋊C8.217C22, C23.19(C4⋊C4), (C4×C8).138C22, C42.122(C2×C4), (C22×C4).249D4, (C4×M4(2)).2C2, C4.46(C2×M4(2)), C4.138(C8⋊C22), (C2×M4(2)).11C4, C4.132(C8.C22), C4⋊M4(2).22C2, C2.9(C4⋊M4(2)), C42.6C4.29C2, (C2×C42).225C22, C2.4(M4(2).C4), C2.4(M4(2)⋊C4), (C2×C4).33(C4⋊C4), (C2×C8).122(C2×C4), C22.83(C2×C4⋊C4), (C2×C4).153(C2×Q8), (C2×C4).1462(C2×D4), (C2×C4).508(C22×C4), (C22×C4).247(C2×C4), SmallGroup(128,301)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C81M4(2)
C1C2C22C2×C4C42C2×C42C4×M4(2) — C81M4(2)
C1C2C2×C4 — C81M4(2)
C1C2×C4C2×C42 — C81M4(2)
C1C22C22C42 — C81M4(2)

Generators and relations for C81M4(2)
 G = < a,b,c | a8=b8=c2=1, bab-1=a3, cac=a5, cbc=b5 >

Subgroups: 132 in 83 conjugacy classes, 50 normal (30 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×3], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×5], C23, C42 [×4], C2×C8 [×4], C2×C8 [×4], M4(2) [×6], C22×C4 [×3], C4×C8 [×2], C8⋊C4 [×2], C8⋊C4, C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C2×M4(2) [×2], C2×M4(2), C82C8 [×2], C81C8 [×2], C4×M4(2), C4⋊M4(2), C42.6C4, C81M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, C2×C4⋊C4, C2×M4(2) [×2], C8⋊C22, C8.C22, C4⋊M4(2), M4(2)⋊C4, M4(2).C4, C81M4(2)

Smallest permutation representation of C81M4(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 16 48 26 39 60 54 19)(2 11 41 29 40 63 55 22)(3 14 42 32 33 58 56 17)(4 9 43 27 34 61 49 20)(5 12 44 30 35 64 50 23)(6 15 45 25 36 59 51 18)(7 10 46 28 37 62 52 21)(8 13 47 31 38 57 53 24)
(2 6)(4 8)(9 57)(10 62)(11 59)(12 64)(13 61)(14 58)(15 63)(16 60)(17 32)(18 29)(19 26)(20 31)(21 28)(22 25)(23 30)(24 27)(34 38)(36 40)(41 45)(43 47)(49 53)(51 55)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K8A···8H8I···8P
order12222444444444448···88···8
size11114111122224444···48···8

32 irreducible representations

dim1111111122222444
type+++++++-+-+-
imageC1C2C2C2C2C2C4C4D4Q8D4Q8M4(2)C8⋊C22C8.C22M4(2).C4
kernelC81M4(2)C82C8C81C8C4×M4(2)C4⋊M4(2)C42.6C4C8⋊C4C2×M4(2)C42C42C22×C4C22×C4C8C4C4C2
# reps1221114411118112

Matrix representation of C81M4(2) in GL6(𝔽17)

1300000
040000
004400
000004
0001300
009131313
,
020000
200000
0012606
0000611
00501111
00551111
,
100000
0160000
001011
000100
0000160
0000016

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,9,0,0,4,0,13,13,0,0,0,0,0,13,0,0,0,4,0,13],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,12,0,5,5,0,0,6,0,0,5,0,0,0,6,11,11,0,0,6,11,11,11],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,1,0,16,0,0,0,1,0,0,16] >;

C81M4(2) in GAP, Magma, Sage, TeX

C_8\rtimes_1M_4(2)
% in TeX

G:=Group("C8:1M4(2)");
// GroupNames label

G:=SmallGroup(128,301);
// by ID

G=gap.SmallGroup(128,301);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,64,1430,387,1123,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a^5,c*b*c=b^5>;
// generators/relations

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