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G = C82⋊C2order 128 = 27

1st semidirect product of C82 and C2 acting faithfully

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

Aliases: C821C2, C23.25C42, C42.742C23, (C2×C8)⋊9C8, C84(C4⋊C8), C4.4(C4×C8), C4⋊C8.26C4, (C4×C8).34C4, C8.23(C2×C8), C82(C8⋊C8), C8⋊C825C2, C83(C22⋊C8), C22.4(C4×C8), C4.52(C8○D4), C22⋊C8.24C4, C4.33(C22×C8), (C22×C8).43C4, (C2×C4).59C42, (C4×C8).359C22, C42.292(C2×C4), C22.22(C2×C42), C82(C42.12C4), C2.2(C82M4(2)), C42.12C4.50C2, (C2×C42).1029C22, C2.3(C2×C4×C8), (C4×C8)(C4⋊C8), (C2×C4×C8).58C2, (C4×C8)(C22⋊C8), (C2×C4).79(C2×C8), (C2×C8)2(C8⋊C8), (C2×C8).263(C2×C4), (C2×C4).579(C22×C4), (C22×C4).374(C2×C4), (C2×C8)2(C42.12C4), SmallGroup(128,182)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C82⋊C2
C1C2C22C2×C4C42C2×C42C42.12C4 — C82⋊C2
C1C2 — C82⋊C2
C1C4×C8 — C82⋊C2
C1C22C22C42 — C82⋊C2

Generators and relations for C82⋊C2
 G = < a,b,c | a8=b8=c2=1, ab=ba, ac=ca, cbc=a4b >

Subgroups: 124 in 106 conjugacy classes, 88 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2×C4, C2×C4, C2×C4, C23, C42, C42, C2×C8, C22×C4, C22×C4, C4×C8, C4×C8, C22⋊C8, C4⋊C8, C2×C42, C22×C8, C82, C8⋊C8, C2×C4×C8, C42.12C4, C82⋊C2
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, C4×C8, C2×C42, C22×C8, C8○D4, C2×C4×C8, C82M4(2), C82⋊C2

Smallest permutation representation of C82⋊C2
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 40 55 20 10 57 25)(2 43 33 56 21 11 58 26)(3 44 34 49 22 12 59 27)(4 45 35 50 23 13 60 28)(5 46 36 51 24 14 61 29)(6 47 37 52 17 15 62 30)(7 48 38 53 18 16 63 31)(8 41 39 54 19 9 64 32)
(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4R8A···8P8Q···8BD
order1222224···44···48···88···8
size1111221···12···21···12···2

80 irreducible representations

dim11111111112
type+++++
imageC1C2C2C2C2C4C4C4C4C8C8○D4
kernelC82⋊C2C82C8⋊C8C2×C4×C8C42.12C4C4×C8C22⋊C8C4⋊C8C22×C8C2×C8C4
# reps1221248843216

Matrix representation of C82⋊C2 in GL3(𝔽17) generated by

1600
0150
0015
,
800
001
0160
,
1600
010
0016
G:=sub<GL(3,GF(17))| [16,0,0,0,15,0,0,0,15],[8,0,0,0,0,16,0,1,0],[16,0,0,0,1,0,0,0,16] >;

C82⋊C2 in GAP, Magma, Sage, TeX

C_8^2\rtimes C_2
% in TeX

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

G:=SmallGroup(128,182);
// by ID

G=gap.SmallGroup(128,182);
# by ID

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

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

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