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G = (C2×C6)⋊8D8order 192 = 26·3

2nd semidirect product of C2×C6 and D8 acting via D8/D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C6)⋊8D8, (C3×D4)⋊13D4, C6.72(C2×D8), D45(C3⋊D4), C35(C22⋊D8), (C22×D4)⋊4S3, C127D425C2, C6.71C22≀C2, C224(D4⋊S3), (C2×D4).198D6, C12.205(C2×D4), (C2×C12).300D4, D4⋊Dic338C2, (C2×D12)⋊14C22, C4⋊Dic321C22, (C22×C6).196D4, (C22×C4).165D6, C12.55D415C2, C6.102(C8⋊C22), (C2×C12).472C23, C2.4(C244S3), (C6×D4).240C22, C23.92(C3⋊D4), C2.22(D126C22), (C22×C12).197C22, (D4×C2×C6)⋊1C2, (C2×D4⋊S3)⋊23C2, (C2×C3⋊C8)⋊10C22, C2.26(C2×D4⋊S3), C4.58(C2×C3⋊D4), (C2×C6).553(C2×D4), (C2×C4).83(C3⋊D4), (C2×C4).558(C22×S3), C22.216(C2×C3⋊D4), SmallGroup(192,776)

Series: Derived Chief Lower central Upper central

C1C2×C12 — (C2×C6)⋊8D8
C1C3C6C2×C6C2×C12C2×D12C2×D4⋊S3 — (C2×C6)⋊8D8
C3C6C2×C12 — (C2×C6)⋊8D8
C1C22C22×C4C22×D4

Generators and relations for (C2×C6)⋊8D8
 G = < a,b,c,d | a2=b6=c8=d2=1, ab=ba, cac-1=dad=ab3, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 584 in 198 conjugacy classes, 51 normal (25 characteristic)
C1, C2 [×3], C2 [×7], C3, C4 [×2], C4 [×2], C22, C22 [×2], C22 [×21], S3, C6 [×3], C6 [×6], C8 [×2], C2×C4 [×2], C2×C4 [×3], D4 [×4], D4 [×10], C23, C23 [×11], Dic3, C12 [×2], C12, D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×18], C22⋊C4, C4⋊C4, C2×C8 [×2], D8 [×4], C22×C4, C2×D4 [×2], C2×D4 [×7], C24, C3⋊C8 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C3×D4 [×6], C22×S3, C22×C6, C22×C6 [×10], C22⋊C8, D4⋊C4 [×2], C4⋊D4, C2×D8 [×2], C22×D4, C2×C3⋊C8 [×2], C4⋊Dic3, D6⋊C4, D4⋊S3 [×4], C2×D12, C2×C3⋊D4, C22×C12, C6×D4 [×2], C6×D4 [×5], C23×C6, C22⋊D8, C12.55D4, D4⋊Dic3 [×2], C127D4, C2×D4⋊S3 [×2], D4×C2×C6, (C2×C6)⋊8D8
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], D8 [×2], C2×D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C2×D8, C8⋊C22, D4⋊S3 [×2], C2×C3⋊D4 [×3], C22⋊D8, C2×D4⋊S3, D126C22, C244S3, (C2×C6)⋊8D8

Smallest permutation representation of (C2×C6)⋊8D8
On 48 points
Generators in S48
(2 24)(4 18)(6 20)(8 22)(10 28)(12 30)(14 32)(16 26)(34 48)(36 42)(38 44)(40 46)
(1 13 45 23 31 39)(2 40 32 24 46 14)(3 15 47 17 25 33)(4 34 26 18 48 16)(5 9 41 19 27 35)(6 36 28 20 42 10)(7 11 43 21 29 37)(8 38 30 22 44 12)
(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)
(1 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 24)(8 23)(9 26)(10 25)(11 32)(12 31)(13 30)(14 29)(15 28)(16 27)(33 42)(34 41)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)

G:=sub<Sym(48)| (2,24)(4,18)(6,20)(8,22)(10,28)(12,30)(14,32)(16,26)(34,48)(36,42)(38,44)(40,46), (1,13,45,23,31,39)(2,40,32,24,46,14)(3,15,47,17,25,33)(4,34,26,18,48,16)(5,9,41,19,27,35)(6,36,28,20,42,10)(7,11,43,21,29,37)(8,38,30,22,44,12), (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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,24)(8,23)(9,26)(10,25)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(33,42)(34,41)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)>;

G:=Group( (2,24)(4,18)(6,20)(8,22)(10,28)(12,30)(14,32)(16,26)(34,48)(36,42)(38,44)(40,46), (1,13,45,23,31,39)(2,40,32,24,46,14)(3,15,47,17,25,33)(4,34,26,18,48,16)(5,9,41,19,27,35)(6,36,28,20,42,10)(7,11,43,21,29,37)(8,38,30,22,44,12), (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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,24)(8,23)(9,26)(10,25)(11,32)(12,31)(13,30)(14,29)(15,28)(16,27)(33,42)(34,41)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43) );

G=PermutationGroup([(2,24),(4,18),(6,20),(8,22),(10,28),(12,30),(14,32),(16,26),(34,48),(36,42),(38,44),(40,46)], [(1,13,45,23,31,39),(2,40,32,24,46,14),(3,15,47,17,25,33),(4,34,26,18,48,16),(5,9,41,19,27,35),(6,36,28,20,42,10),(7,11,43,21,29,37),(8,38,30,22,44,12)], [(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)], [(1,22),(2,21),(3,20),(4,19),(5,18),(6,17),(7,24),(8,23),(9,26),(10,25),(11,32),(12,31),(13,30),(14,29),(15,28),(16,27),(33,42),(34,41),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43)])

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J 3 4A4B4C4D6A···6G6H···6O8A8B8C8D12A12B12C12D
order12222222222344446···66···6888812121212
size1111224444242224242···24···4121212124444

39 irreducible representations

dim1111112222222222444
type+++++++++++++++
imageC1C2C2C2C2C2S3D4D4D4D6D6D8C3⋊D4C3⋊D4C3⋊D4C8⋊C22D4⋊S3D126C22
kernel(C2×C6)⋊8D8C12.55D4D4⋊Dic3C127D4C2×D4⋊S3D4×C2×C6C22×D4C2×C12C3×D4C22×C6C22×C4C2×D4C2×C6C2×C4D4C23C6C22C2
# reps1121211141124282122

Matrix representation of (C2×C6)⋊8D8 in GL4(𝔽73) generated by

1000
07200
0010
0001
,
65000
0900
0010
0001
,
07200
72000
005716
005757
,
0100
1000
005716
001616
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[65,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,72,0,0,72,0,0,0,0,0,57,57,0,0,16,57],[0,1,0,0,1,0,0,0,0,0,57,16,0,0,16,16] >;

(C2×C6)⋊8D8 in GAP, Magma, Sage, TeX

(C_2\times C_6)\rtimes_8D_8
% in TeX

G:=Group("(C2xC6):8D8");
// GroupNames label

G:=SmallGroup(192,776);
// by ID

G=gap.SmallGroup(192,776);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,1684,851,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^6=c^8=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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