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G = C83M4(2)  order 128 = 27

3rd semidirect product of C8 and M4(2) acting via M4(2)/C4=C22

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

Aliases: C83M4(2), C42.651C23, C8⋊C88C2, D4⋊C836C2, C82C813C2, C86D431C2, (C4×D8).16C2, (C2×D8).12C4, (C2×C8).185D4, C2.D8.21C4, C4.18(C8○D4), D4⋊C4.12C4, C2.5(D8⋊C4), C2.11(C86D4), C4⋊C8.229C22, (C4×C8).320C22, (C4×D4).16C22, C22.142(C4×D4), C4.12(C2×M4(2)), C2.12(C8.26D4), C4.147(C8⋊C22), C4⋊C4.64(C2×C4), (C2×C8).59(C2×C4), (C2×D4).61(C2×C4), (C2×C4).1487(C2×D4), (C2×C4).512(C4○D4), (C2×C4).343(C22×C4), SmallGroup(128,326)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C83M4(2)
C1C2C22C2×C4C42C4×C8C86D4 — C83M4(2)
C1C2C2×C4 — C83M4(2)
C1C2×C4C4×C8 — C83M4(2)
C1C22C22C42 — C83M4(2)

Generators and relations for C83M4(2)
 G = < a,b,c | a8=b8=c2=1, bab-1=a3, cac=a-1, cbc=b5 >

Subgroups: 184 in 88 conjugacy classes, 42 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×6], C8 [×2], C8 [×6], C2×C4 [×3], C2×C4 [×6], D4 [×4], C23 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C2×C8 [×3], M4(2) [×4], D8 [×2], C22×C4 [×2], C2×D4 [×2], C4×C8 [×3], C22⋊C8 [×2], D4⋊C4 [×2], C4⋊C8 [×2], C2.D8, C4×D4 [×2], C2×M4(2) [×2], C2×D8, C8⋊C8, D4⋊C8 [×2], C82C8, C86D4 [×2], C4×D8, C83M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×M4(2), C8○D4, C8⋊C22 [×2], C86D4, D8⋊C4, C8.26D4, C83M4(2)

Smallest permutation representation of C83M4(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 25 63 54 23 39 11 41)(2 28 64 49 24 34 12 44)(3 31 57 52 17 37 13 47)(4 26 58 55 18 40 14 42)(5 29 59 50 19 35 15 45)(6 32 60 53 20 38 16 48)(7 27 61 56 21 33 9 43)(8 30 62 51 22 36 10 46)
(2 8)(3 7)(4 6)(9 13)(10 12)(14 16)(17 21)(18 20)(22 24)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(32 40)(41 54)(42 53)(43 52)(44 51)(45 50)(46 49)(47 56)(48 55)(57 61)(58 60)(62 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), (1,25,63,54,23,39,11,41)(2,28,64,49,24,34,12,44)(3,31,57,52,17,37,13,47)(4,26,58,55,18,40,14,42)(5,29,59,50,19,35,15,45)(6,32,60,53,20,38,16,48)(7,27,61,56,21,33,9,43)(8,30,62,51,22,36,10,46), (2,8)(3,7)(4,6)(9,13)(10,12)(14,16)(17,21)(18,20)(22,24)(25,39)(26,38)(27,37)(28,36)(29,35)(30,34)(31,33)(32,40)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,56)(48,55)(57,61)(58,60)(62,64)>;

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8L8M8N8O8P
order12222244444444448···88888
size11118811112222884···48888

32 irreducible representations

dim111111111222244
type++++++++
imageC1C2C2C2C2C2C4C4C4D4M4(2)C4○D4C8○D4C8⋊C22C8.26D4
kernelC83M4(2)C8⋊C8D4⋊C8C82C8C86D4C4×D8D4⋊C4C2.D8C2×D8C2×C8C8C2×C4C4C4C2
# reps112121422242422

Matrix representation of C83M4(2) in GL6(𝔽17)

0150000
900000
00413611
00151566
0014124
00114213
,
010000
1300000
0001150
0016002
0016001
0001160
,
100000
0160000
001000
0001600
0001610
0010016

G:=sub<GL(6,GF(17))| [0,9,0,0,0,0,15,0,0,0,0,0,0,0,4,15,14,1,0,0,13,15,1,14,0,0,6,6,2,2,0,0,11,6,4,13],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,16,16,0,0,0,1,0,0,1,0,0,15,0,0,16,0,0,0,2,1,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,1,0,0,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

C83M4(2) in GAP, Magma, Sage, TeX

C_8\rtimes_3M_4(2)
% in TeX

G:=Group("C8:3M4(2)");
// GroupNames label

G:=SmallGroup(128,326);
// by ID

G=gap.SmallGroup(128,326);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,2102,723,268,1123,570,136,172]);
// Polycyclic

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

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