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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: C82:1C2, C23.25C42, C42.742C23, (C2xC8):9C8, C8o4(C4:C8), C4.4(C4xC8), C4:C8.26C4, (C4xC8).34C4, C8.23(C2xC8), C8o2(C8:C8), C8:C8:25C2, C8o3(C22:C8), C22.4(C4xC8), C4.52(C8oD4), C22:C8.24C4, C4.33(C22xC8), (C22xC8).43C4, (C2xC4).59C42, (C4xC8).359C22, C42.292(C2xC4), C22.22(C2xC42), C8o2(C42.12C4), C2.2(C8o2M4(2)), C42.12C4.50C2, (C2xC42).1029C22, C2.3(C2xC4xC8), (C4xC8)o(C4:C8), (C2xC4xC8).58C2, (C4xC8)o(C22:C8), (C2xC4).79(C2xC8), (C2xC8)o2(C8:C8), (C2xC8).263(C2xC4), (C2xC4).579(C22xC4), (C22xC4).374(C2xC4), (C2xC8)o2(C42.12C4), SmallGroup(128,182)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C82:C2
C1C2C22C2xC4C42C2xC42C42.12C4 — C82:C2
C1C2 — C82:C2
C1C4xC8 — 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, C2xC4, C2xC4, C2xC4, C23, C42, C42, C2xC8, C22xC4, C22xC4, C4xC8, C4xC8, C22:C8, C4:C8, C2xC42, C22xC8, C82, C8:C8, C2xC4xC8, C42.12C4, C82:C2
Quotients: C1, C2, C4, C22, C8, C2xC4, C23, C42, C2xC8, C22xC4, C4xC8, C2xC42, C22xC8, C8oD4, C2xC4xC8, C8o2M4(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+++++
imageC1C2C2C2C2C4C4C4C4C8C8oD4
kernelC82:C2C82C8:C8C2xC4xC8C42.12C4C4xC8C22:C8C4:C8C22xC8C2xC8C4
# 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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