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G = C8.2C42order 128 = 27

2nd non-split extension by C8 of C42 acting via C42/C22=C22

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C8.2C42, M5(2)⋊9C4, C23.30SD16, (C2×C8).6Q8, C8.13(C4⋊C4), (C2×C4).9Q16, C8.C43C4, (C2×C8).338D4, (C2×C4).123D8, C2.D8.17C4, C4.3(C2.D8), (C2×C4).22SD16, C8.37(C22⋊C4), (C22×C4).193D4, C4.33(D4⋊C4), C22.3(C4.Q8), C4.15(Q8⋊C4), C2.2(M5(2)⋊C2), C2.2(C8.17D4), (C2×M5(2)).15C2, C4.8(C2.C42), (C22×C8).206C22, C22.3(Q8⋊C4), C22.47(D4⋊C4), C2.17(C22.4Q16), (C2×C8).53(C2×C4), (C2×C4).29(C4⋊C4), (C2×C2.D8).31C2, (C2×C8.C4).5C2, (C2×C4).231(C22⋊C4), SmallGroup(128,119)

Series: Derived Chief Lower central Upper central Jennings

C1C8 — C8.2C42
C1C2C4C8C2×C8C22×C8C2×M5(2) — C8.2C42
C1C2C4C8 — C8.2C42
C1C22C22×C4C22×C8 — C8.2C42
C1C2C2C2C2C4C4C22×C8 — C8.2C42

Generators and relations for C8.2C42
 G = < a,b,c | a8=b4=1, c4=a2, bab-1=a-1, cac-1=a5, cbc-1=a3b >

Subgroups: 136 in 68 conjugacy classes, 40 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×4], C23, C16 [×2], C4⋊C4 [×3], C2×C8 [×6], C2×C8, M4(2) [×3], C22×C4, C22×C4, C2.D8 [×2], C2.D8, C8.C4 [×2], C8.C4, C2×C16, M5(2) [×2], M5(2), C2×C4⋊C4, C22×C8, C2×M4(2), C2×C2.D8, C2×C8.C4, C2×M5(2), C8.2C42
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], D8, SD16 [×2], Q16, C2.C42, D4⋊C4 [×2], Q8⋊C4 [×2], C4.Q8, C2.D8, C22.4Q16, M5(2)⋊C2, C8.17D4, C8.2C42

Smallest permutation representation of C8.2C42
On 64 points
Generators in S64
(1 3 5 7 9 11 13 15)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 28 22 32 26 20 30 24)(33 43 37 47 41 35 45 39)(34 36 38 40 42 44 46 48)(49 51 53 55 57 59 61 63)(50 60 54 64 58 52 62 56)
(1 48 19 53)(2 43 20 64)(3 46 21 51)(4 41 22 62)(5 44 23 49)(6 39 24 60)(7 42 25 63)(8 37 26 58)(9 40 27 61)(10 35 28 56)(11 38 29 59)(12 33 30 54)(13 36 31 57)(14 47 32 52)(15 34 17 55)(16 45 18 50)
(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)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H8A8B8C8D8E8F8G8H8I8J16A···16H
order12222244444444888888888816···16
size1111222222888822224488884···4

32 irreducible representations

dim1111111222222244
type+++++-++-+-
imageC1C2C2C2C4C4C4D4Q8D4D8SD16Q16SD16M5(2)⋊C2C8.17D4
kernelC8.2C42C2×C2.D8C2×C8.C4C2×M5(2)C2.D8C8.C4M5(2)C2×C8C2×C8C22×C4C2×C4C2×C4C2×C4C23C2C2
# reps1111444211222222

Matrix representation of C8.2C42 in GL6(𝔽17)

1600000
0160000
0031400
003300
0000143
00001414
,
690000
11110000
003300
0031400
000010
0000016
,
690000
15110000
000010
000001
0031400
003300

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,0,0,0,0,14,14,0,0,0,0,3,14],[6,11,0,0,0,0,9,11,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[6,15,0,0,0,0,9,11,0,0,0,0,0,0,0,0,3,3,0,0,0,0,14,3,0,0,1,0,0,0,0,0,0,1,0,0] >;

C8.2C42 in GAP, Magma, Sage, TeX

C_8._2C_4^2
% in TeX

G:=Group("C8.2C4^2");
// GroupNames label

G:=SmallGroup(128,119);
// by ID

G=gap.SmallGroup(128,119);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,723,520,1684,242,4037,124]);
// Polycyclic

G:=Group<a,b,c|a^8=b^4=1,c^4=a^2,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b>;
// generators/relations

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