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## G = (C6×Q8)⋊6C4order 192 = 26·3

### 2nd semidirect product of C6×Q8 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — (C6×Q8)⋊6C4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4⋊Dic3 — C23.26D6 — (C6×Q8)⋊6C4
 Lower central C3 — C6 — C12 — (C6×Q8)⋊6C4
 Upper central C1 — C22 — C22×C4 — C22×Q8

Generators and relations for (C6×Q8)⋊6C4
G = < a,b,c,d | a6=b4=d4=1, c2=b2, ab=ba, ac=ca, dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=a3b-1c >

Subgroups: 296 in 150 conjugacy classes, 71 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×8], Q8 [×4], Q8 [×6], C23, Dic3 [×2], C12 [×2], C12 [×2], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4, C2×Q8 [×6], C2×Q8 [×3], C3⋊C8 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×4], C2×C12 [×6], C3×Q8 [×4], C3×Q8 [×6], C22×C6, Q8⋊C4 [×4], C42⋊C2, C2×M4(2), C22×Q8, C2×C3⋊C8 [×2], C4.Dic3 [×2], C4×Dic3, C4⋊Dic3 [×2], C6.D4, C22×C12, C22×C12, C6×Q8 [×6], C6×Q8 [×3], C23.38D4, Q82Dic3 [×4], C2×C4.Dic3, C23.26D6, Q8×C2×C6, (C6×Q8)⋊6C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C8.C22 [×2], C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C23.38D4, Q8.11D6 [×2], C2×C6.D4, (C6×Q8)⋊6C4

Smallest permutation representation of (C6×Q8)⋊6C4
On 96 points
Generators in S96
(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 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 28 9 20)(2 29 10 21)(3 30 11 22)(4 25 12 23)(5 26 7 24)(6 27 8 19)(13 82 90 94)(14 83 85 95)(15 84 86 96)(16 79 87 91)(17 80 88 92)(18 81 89 93)(31 38 54 47)(32 39 49 48)(33 40 50 43)(34 41 51 44)(35 42 52 45)(36 37 53 46)(55 74 67 63)(56 75 68 64)(57 76 69 65)(58 77 70 66)(59 78 71 61)(60 73 72 62)
(1 44 9 41)(2 45 10 42)(3 46 11 37)(4 47 12 38)(5 48 7 39)(6 43 8 40)(13 58 90 70)(14 59 85 71)(15 60 86 72)(16 55 87 67)(17 56 88 68)(18 57 89 69)(19 33 27 50)(20 34 28 51)(21 35 29 52)(22 36 30 53)(23 31 25 54)(24 32 26 49)(61 95 78 83)(62 96 73 84)(63 91 74 79)(64 92 75 80)(65 93 76 81)(66 94 77 82)
(1 58 12 55)(2 69 7 72)(3 56 8 59)(4 67 9 70)(5 60 10 57)(6 71 11 68)(13 34 16 54)(14 50 17 36)(15 32 18 52)(19 78 30 75)(20 66 25 63)(21 76 26 73)(22 64 27 61)(23 74 28 77)(24 62 29 65)(31 90 51 87)(33 88 53 85)(35 86 49 89)(37 83 43 80)(38 94 44 91)(39 81 45 84)(40 92 46 95)(41 79 47 82)(42 96 48 93)

G:=sub<Sym(96)| (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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,28,9,20)(2,29,10,21)(3,30,11,22)(4,25,12,23)(5,26,7,24)(6,27,8,19)(13,82,90,94)(14,83,85,95)(15,84,86,96)(16,79,87,91)(17,80,88,92)(18,81,89,93)(31,38,54,47)(32,39,49,48)(33,40,50,43)(34,41,51,44)(35,42,52,45)(36,37,53,46)(55,74,67,63)(56,75,68,64)(57,76,69,65)(58,77,70,66)(59,78,71,61)(60,73,72,62), (1,44,9,41)(2,45,10,42)(3,46,11,37)(4,47,12,38)(5,48,7,39)(6,43,8,40)(13,58,90,70)(14,59,85,71)(15,60,86,72)(16,55,87,67)(17,56,88,68)(18,57,89,69)(19,33,27,50)(20,34,28,51)(21,35,29,52)(22,36,30,53)(23,31,25,54)(24,32,26,49)(61,95,78,83)(62,96,73,84)(63,91,74,79)(64,92,75,80)(65,93,76,81)(66,94,77,82), (1,58,12,55)(2,69,7,72)(3,56,8,59)(4,67,9,70)(5,60,10,57)(6,71,11,68)(13,34,16,54)(14,50,17,36)(15,32,18,52)(19,78,30,75)(20,66,25,63)(21,76,26,73)(22,64,27,61)(23,74,28,77)(24,62,29,65)(31,90,51,87)(33,88,53,85)(35,86,49,89)(37,83,43,80)(38,94,44,91)(39,81,45,84)(40,92,46,95)(41,79,47,82)(42,96,48,93)>;

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)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,28,9,20)(2,29,10,21)(3,30,11,22)(4,25,12,23)(5,26,7,24)(6,27,8,19)(13,82,90,94)(14,83,85,95)(15,84,86,96)(16,79,87,91)(17,80,88,92)(18,81,89,93)(31,38,54,47)(32,39,49,48)(33,40,50,43)(34,41,51,44)(35,42,52,45)(36,37,53,46)(55,74,67,63)(56,75,68,64)(57,76,69,65)(58,77,70,66)(59,78,71,61)(60,73,72,62), (1,44,9,41)(2,45,10,42)(3,46,11,37)(4,47,12,38)(5,48,7,39)(6,43,8,40)(13,58,90,70)(14,59,85,71)(15,60,86,72)(16,55,87,67)(17,56,88,68)(18,57,89,69)(19,33,27,50)(20,34,28,51)(21,35,29,52)(22,36,30,53)(23,31,25,54)(24,32,26,49)(61,95,78,83)(62,96,73,84)(63,91,74,79)(64,92,75,80)(65,93,76,81)(66,94,77,82), (1,58,12,55)(2,69,7,72)(3,56,8,59)(4,67,9,70)(5,60,10,57)(6,71,11,68)(13,34,16,54)(14,50,17,36)(15,32,18,52)(19,78,30,75)(20,66,25,63)(21,76,26,73)(22,64,27,61)(23,74,28,77)(24,62,29,65)(31,90,51,87)(33,88,53,85)(35,86,49,89)(37,83,43,80)(38,94,44,91)(39,81,45,84)(40,92,46,95)(41,79,47,82)(42,96,48,93) );

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),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,28,9,20),(2,29,10,21),(3,30,11,22),(4,25,12,23),(5,26,7,24),(6,27,8,19),(13,82,90,94),(14,83,85,95),(15,84,86,96),(16,79,87,91),(17,80,88,92),(18,81,89,93),(31,38,54,47),(32,39,49,48),(33,40,50,43),(34,41,51,44),(35,42,52,45),(36,37,53,46),(55,74,67,63),(56,75,68,64),(57,76,69,65),(58,77,70,66),(59,78,71,61),(60,73,72,62)], [(1,44,9,41),(2,45,10,42),(3,46,11,37),(4,47,12,38),(5,48,7,39),(6,43,8,40),(13,58,90,70),(14,59,85,71),(15,60,86,72),(16,55,87,67),(17,56,88,68),(18,57,89,69),(19,33,27,50),(20,34,28,51),(21,35,29,52),(22,36,30,53),(23,31,25,54),(24,32,26,49),(61,95,78,83),(62,96,73,84),(63,91,74,79),(64,92,75,80),(65,93,76,81),(66,94,77,82)], [(1,58,12,55),(2,69,7,72),(3,56,8,59),(4,67,9,70),(5,60,10,57),(6,71,11,68),(13,34,16,54),(14,50,17,36),(15,32,18,52),(19,78,30,75),(20,66,25,63),(21,76,26,73),(22,64,27,61),(23,74,28,77),(24,62,29,65),(31,90,51,87),(33,88,53,85),(35,86,49,89),(37,83,43,80),(38,94,44,91),(39,81,45,84),(40,92,46,95),(41,79,47,82),(42,96,48,93)])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A ··· 6G 8A 8B 8C 8D 12A ··· 12L order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 8 8 8 8 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 12 12 12 12 2 ··· 2 12 12 12 12 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + - image C1 C2 C2 C2 C2 C4 S3 D4 D4 D6 Dic3 D6 C3⋊D4 C3⋊D4 C8.C22 Q8.11D6 kernel (C6×Q8)⋊6C4 Q8⋊2Dic3 C2×C4.Dic3 C23.26D6 Q8×C2×C6 C6×Q8 C22×Q8 C2×C12 C22×C6 C22×C4 C2×Q8 C2×Q8 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 8 1 3 1 1 4 2 6 2 2 4

Matrix representation of (C6×Q8)⋊6C4 in GL6(𝔽73)

 72 1 0 0 0 0 72 0 0 0 0 0 0 0 65 0 0 0 0 0 0 65 0 0 0 0 49 16 64 0 0 0 41 1 0 64
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 16 48 0 1 0 0 48 57 72 0
,
 30 13 0 0 0 0 60 43 0 0 0 0 0 0 11 43 0 0 0 0 43 62 0 0 0 0 36 53 30 11 0 0 17 0 11 43
,
 11 25 0 0 0 0 36 62 0 0 0 0 0 0 22 34 7 0 0 0 5 66 0 7 0 0 11 21 51 39 0 0 31 21 68 7

G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,1,0,0,0,0,0,0,0,65,0,49,41,0,0,0,65,16,1,0,0,0,0,64,0,0,0,0,0,0,64],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,16,48,0,0,1,0,48,57,0,0,0,0,0,72,0,0,0,0,1,0],[30,60,0,0,0,0,13,43,0,0,0,0,0,0,11,43,36,17,0,0,43,62,53,0,0,0,0,0,30,11,0,0,0,0,11,43],[11,36,0,0,0,0,25,62,0,0,0,0,0,0,22,5,11,31,0,0,34,66,21,21,0,0,7,0,51,68,0,0,0,7,39,7] >;

(C6×Q8)⋊6C4 in GAP, Magma, Sage, TeX

(C_6\times Q_8)\rtimes_6C_4
% in TeX

G:=Group("(C6xQ8):6C4");
// GroupNames label

G:=SmallGroup(192,784);
// by ID

G=gap.SmallGroup(192,784);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,422,387,184,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^3*b^-1*c>;
// generators/relations

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