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## G = (C2×C6)⋊8D8order 192 = 26·3

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — (C2×C6)⋊8D8
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C2×D4⋊S3 — (C2×C6)⋊8D8
 Lower central C3 — C6 — C2×C12 — (C2×C6)⋊8D8
 Upper central C1 — C22 — C22×C4 — C22×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 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 3 4A 4B 4C 4D 6A ··· 6G 6H ··· 6O 8A 8B 8C 8D 12A 12B 12C 12D order 1 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 6 ··· 6 6 ··· 6 8 8 8 8 12 12 12 12 size 1 1 1 1 2 2 4 4 4 4 24 2 2 2 4 24 2 ··· 2 4 ··· 4 12 12 12 12 4 4 4 4

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 D8 C3⋊D4 C3⋊D4 C3⋊D4 C8⋊C22 D4⋊S3 D12⋊6C22 kernel (C2×C6)⋊8D8 C12.55D4 D4⋊Dic3 C12⋊7D4 C2×D4⋊S3 D4×C2×C6 C22×D4 C2×C12 C3×D4 C22×C6 C22×C4 C2×D4 C2×C6 C2×C4 D4 C23 C6 C22 C2 # reps 1 1 2 1 2 1 1 1 4 1 1 2 4 2 8 2 1 2 2

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

 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 16 0 0 57 57
,
 0 1 0 0 1 0 0 0 0 0 57 16 0 0 16 16
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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