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G = C84D8order 128 = 27

1st semidirect product of C8 and D8 acting via D8/C8=C2

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

Aliases: C84D8, C826C2, C42.659C23, C4.3(C2×D8), C84D44C2, (C2×C8).241D4, C2.6(C84D4), (C4×C8).394C22, C41D4.45C22, C22.60(C41D4), (C2×C4).716(C2×D4), SmallGroup(128,444)

Series: Derived Chief Lower central Upper central Jennings

C1C42 — C84D8
C1C2C22C2×C4C42C4×C8C82 — C84D8
C1C22C42 — C84D8
C1C22C42 — C84D8
C1C22C22C42 — C84D8

Generators and relations for C84D8
 G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 464 in 140 conjugacy classes, 48 normal (5 characteristic)
C1, C2 [×3], C2 [×4], C4 [×6], C22, C22 [×12], C8 [×12], C2×C4 [×3], D4 [×24], C23 [×4], C42, C2×C8 [×6], D8 [×24], C2×D4 [×12], C4×C8 [×3], C41D4 [×4], C2×D8 [×12], C82, C84D4 [×6], C84D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×12], C2×D4 [×3], C41D4, C2×D8 [×6], C84D4 [×3], C84D8

Smallest permutation representation of C84D8
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 39 18 25 12 45 61 49)(2 40 19 26 13 46 62 50)(3 33 20 27 14 47 63 51)(4 34 21 28 15 48 64 52)(5 35 22 29 16 41 57 53)(6 36 23 30 9 42 58 54)(7 37 24 31 10 43 59 55)(8 38 17 32 11 44 60 56)
(1 61)(2 60)(3 59)(4 58)(5 57)(6 64)(7 63)(8 62)(9 21)(10 20)(11 19)(12 18)(13 17)(14 24)(15 23)(16 22)(26 32)(27 31)(28 30)(33 43)(34 42)(35 41)(36 48)(37 47)(38 46)(39 45)(40 44)(50 56)(51 55)(52 54)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F8A···8X
order122222224···48···8
size1111161616162···22···2

38 irreducible representations

dim11122
type+++++
imageC1C2C2D4D8
kernelC84D8C82C84D4C2×C8C8
# reps116624

Matrix representation of C84D8 in GL4(𝔽17) generated by

14300
141400
0010
0001
,
16000
01600
00011
00311
,
1000
01600
00162
0001
G:=sub<GL(4,GF(17))| [14,14,0,0,3,14,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,0,3,0,0,11,11],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,2,1] >;

C84D8 in GAP, Magma, Sage, TeX

C_8\rtimes_4D_8
% in TeX

G:=Group("C8:4D8");
// GroupNames label

G:=SmallGroup(128,444);
// by ID

G=gap.SmallGroup(128,444);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,141,288,422,436,1123,136,2804,172]);
// Polycyclic

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

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