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G = C822C2order 128 = 27

2nd semidirect product of C82 and C2 acting faithfully

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

Aliases: C822C2, C23.13C42, C42.746C23, C4⋊C8.21C4, C8⋊C822C2, C4.56(C8○D4), C22⋊C8.20C4, (C2×C4).17C42, C42.49(C2×C4), (C4×C8).307C22, C22.43(C2×C42), C2.9(C82M4(2)), (C2×C42).143C22, C42.12C4.44C2, (C2×C8).120(C2×C4), (C2×C4).583(C22×C4), (C22×C4).173(C2×C4), SmallGroup(128,186)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C822C2
C1C2C22C2×C4C42C2×C42C42.12C4 — C822C2
C1C22 — C822C2
C1C42 — C822C2
C1C22C22C42 — C822C2

Generators and relations for C822C2
 G = < a,b,c | a8=b8=c2=1, ab=ba, cac=a5b4, cbc=a4b >

Subgroups: 116 in 89 conjugacy classes, 64 normal (7 characteristic)
C1, C2 [×3], C2, C4 [×6], C4 [×3], C22, C22 [×3], C8 [×12], C2×C4 [×6], C2×C4 [×6], C23, C42, C42 [×3], C2×C8 [×12], C22×C4 [×3], C4×C8 [×6], C22⋊C8 [×6], C4⋊C8 [×6], C2×C42, C82, C8⋊C8 [×3], C42.12C4 [×3], C822C2
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], C22×C4 [×3], C2×C42, C8○D4 [×6], C82M4(2) [×3], C822C2

Smallest permutation representation of C822C2
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 42 10 27 55 34 57 19)(2 43 11 28 56 35 58 20)(3 44 12 29 49 36 59 21)(4 45 13 30 50 37 60 22)(5 46 14 31 51 38 61 23)(6 47 15 32 52 39 62 24)(7 48 16 25 53 40 63 17)(8 41 9 26 54 33 64 18)
(2 52)(4 54)(6 56)(8 50)(9 60)(11 62)(13 64)(15 58)(17 21)(18 26)(19 23)(20 28)(22 30)(24 32)(25 29)(27 31)(33 41)(34 38)(35 43)(36 40)(37 45)(39 47)(42 46)(44 48)

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

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

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

56 conjugacy classes

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

56 irreducible representations

dim1111112
type++++
imageC1C2C2C2C4C4C8○D4
kernelC822C2C82C8⋊C8C42.12C4C22⋊C8C4⋊C8C4
# reps1133121224

Matrix representation of C822C2 in GL4(𝔽17) generated by

11600
21600
0080
0008
,
8000
0800
00151
0092
,
11600
01600
0010
00416
G:=sub<GL(4,GF(17))| [1,2,0,0,16,16,0,0,0,0,8,0,0,0,0,8],[8,0,0,0,0,8,0,0,0,0,15,9,0,0,1,2],[1,0,0,0,16,16,0,0,0,0,1,4,0,0,0,16] >;

C822C2 in GAP, Magma, Sage, TeX

C_8^2\rtimes_2C_2
% in TeX

G:=Group("C8^2:2C2");
// GroupNames label

G:=SmallGroup(128,186);
// by ID

G=gap.SmallGroup(128,186);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,56,120,2102,387,136,172]);
// Polycyclic

G:=Group<a,b,c|a^8=b^8=c^2=1,a*b=b*a,c*a*c=a^5*b^4,c*b*c=a^4*b>;
// generators/relations

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