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G = C86M4(2)  order 128 = 27

3rd semidirect product of C8 and M4(2) acting via M4(2)/C8=C2

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

Aliases: C8216C2, C86M4(2), C23.14C42, C42.747C23, C8⋊C823C2, C8⋊C4.13C4, (C2×C4).18C42, C42.50(C2×C4), C2.7(C4×M4(2)), (C4×C8).308C22, C4.60(C2×M4(2)), (C2×M4(2)).23C4, (C4×M4(2)).15C2, C22.44(C2×C42), (C2×C42).144C22, (C2×C8).121(C2×C4), (C22×C4).174(C2×C4), (C2×C4).584(C22×C4), SmallGroup(128,187)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C86M4(2)
C1C2C22C2×C4C42C2×C42C4×M4(2) — C86M4(2)
C1C22 — C86M4(2)
C1C42 — C86M4(2)
C1C22C22C42 — C86M4(2)

Generators and relations for C86M4(2)
 G = < a,b,c | a8=b8=c2=1, bab-1=cac=a5, cbc=b5 >

Subgroups: 140 in 107 conjugacy classes, 76 normal (7 characteristic)
C1, C2 [×3], C2, C4 [×6], C4 [×3], C22, C22 [×3], C8 [×12], C8 [×6], C2×C4 [×6], C2×C4 [×6], C23, C42, C42 [×3], C2×C8 [×12], M4(2) [×12], C22×C4 [×3], C4×C8 [×6], C8⋊C4 [×6], C2×C42, C2×M4(2) [×6], C82, C8⋊C8 [×3], C4×M4(2) [×3], C86M4(2)
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×12], C22×C4 [×3], C2×C42, C2×M4(2) [×6], C4×M4(2) [×3], C86M4(2)

Smallest permutation representation of C86M4(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 10 57 55 20 42 40 25)(2 15 58 52 21 47 33 30)(3 12 59 49 22 44 34 27)(4 9 60 54 23 41 35 32)(5 14 61 51 24 46 36 29)(6 11 62 56 17 43 37 26)(7 16 63 53 18 48 38 31)(8 13 64 50 19 45 39 28)
(2 6)(4 8)(9 45)(10 42)(11 47)(12 44)(13 41)(14 46)(15 43)(16 48)(17 21)(19 23)(25 55)(26 52)(27 49)(28 54)(29 51)(30 56)(31 53)(32 50)(33 37)(35 39)(58 62)(60 64)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D4A···4L4M4N4O8A···8X8Y···8AJ
order122224···44448···88···8
size111141···14442···24···4

56 irreducible representations

dim1111112
type++++
imageC1C2C2C2C4C4M4(2)
kernelC86M4(2)C82C8⋊C8C4×M4(2)C8⋊C4C2×M4(2)C8
# reps1133121224

Matrix representation of C86M4(2) in GL4(𝔽17) generated by

0100
4000
0010
0001
,
13000
0400
0016
00916
,
1000
01600
00113
00016
G:=sub<GL(4,GF(17))| [0,4,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[13,0,0,0,0,4,0,0,0,0,1,9,0,0,6,16],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,13,16] >;

C86M4(2) in GAP, Magma, Sage, TeX

C_8\rtimes_6M_4(2)
% in TeX

G:=Group("C8:6M4(2)");
// GroupNames label

G:=SmallGroup(128,187);
// by ID

G=gap.SmallGroup(128,187);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,120,758,723,136,172]);
// Polycyclic

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

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