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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 [×2], C2 [×2], C4 [×8], C4 [×2], C22, C22 [×2], C22 [×2], C8 [×8], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×4], C23, C42 [×2], C42 [×2], C2×C8 [×20], C22×C4, C22×C4 [×2], C4×C8 [×2], C4×C8 [×6], C22⋊C8 [×4], C4⋊C8 [×4], C2×C42, C22×C8 [×2], C82 [×2], C8⋊C8 [×2], C2×C4×C8, C42.12C4 [×2], C82⋊C2
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], C4×C8 [×4], C2×C42, C22×C8 [×2], C8○D4 [×4], C2×C4×C8, C82M4(2) [×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 34 12 57 29 55 17)(2 43 35 13 58 30 56 18)(3 44 36 14 59 31 49 19)(4 45 37 15 60 32 50 20)(5 46 38 16 61 25 51 21)(6 47 39 9 62 26 52 22)(7 48 40 10 63 27 53 23)(8 41 33 11 64 28 54 24)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(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), (1,42,34,12,57,29,55,17)(2,43,35,13,58,30,56,18)(3,44,36,14,59,31,49,19)(4,45,37,15,60,32,50,20)(5,46,38,16,61,25,51,21)(6,47,39,9,62,26,52,22)(7,48,40,10,63,27,53,23)(8,41,33,11,64,28,54,24), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(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), (1,42,34,12,57,29,55,17)(2,43,35,13,58,30,56,18)(3,44,36,14,59,31,49,19)(4,45,37,15,60,32,50,20)(5,46,38,16,61,25,51,21)(6,47,39,9,62,26,52,22)(7,48,40,10,63,27,53,23)(8,41,33,11,64,28,54,24), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(41,45)(42,46)(43,47)(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)], [(1,42,34,12,57,29,55,17),(2,43,35,13,58,30,56,18),(3,44,36,14,59,31,49,19),(4,45,37,15,60,32,50,20),(5,46,38,16,61,25,51,21),(6,47,39,9,62,26,52,22),(7,48,40,10,63,27,53,23),(8,41,33,11,64,28,54,24)], [(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(41,45),(42,46),(43,47),(44,48)])

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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