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## G = Q8×C2×C6order 96 = 25·3

### Direct product of C2×C6 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C2×C6, C6.17C24, C12.50C23, C4.7(C22×C6), C2.2(C23×C6), (C22×C4).9C6, (C2×C6).84C23, C23.17(C2×C6), (C22×C12).17C2, C22.9(C22×C6), (C2×C12).133C22, (C22×C6).50C22, (C2×C4).29(C2×C6), SmallGroup(96,222)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C2×C6
 Chief series C1 — C2 — C6 — C12 — C3×Q8 — C6×Q8 — Q8×C2×C6
 Lower central C1 — C2 — Q8×C2×C6
 Upper central C1 — C22×C6 — Q8×C2×C6

Generators and relations for Q8×C2×C6
G = < a,b,c,d | a2=b6=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 156, all normal (8 characteristic)
C1, C2, C2 [×6], C3, C4 [×12], C22 [×7], C6, C6 [×6], C2×C4 [×18], Q8 [×16], C23, C12 [×12], C2×C6 [×7], C22×C4 [×3], C2×Q8 [×12], C2×C12 [×18], C3×Q8 [×16], C22×C6, C22×Q8, C22×C12 [×3], C6×Q8 [×12], Q8×C2×C6
Quotients: C1, C2 [×15], C3, C22 [×35], C6 [×15], Q8 [×4], C23 [×15], C2×C6 [×35], C2×Q8 [×6], C24, C3×Q8 [×4], C22×C6 [×15], C22×Q8, C6×Q8 [×6], C23×C6, Q8×C2×C6

Smallest permutation representation of Q8×C2×C6
Regular action on 96 points
Generators in S96
(1 30)(2 25)(3 26)(4 27)(5 28)(6 29)(7 84)(8 79)(9 80)(10 81)(11 82)(12 83)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(31 49)(32 50)(33 51)(34 52)(35 53)(36 54)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(85 91)(86 92)(87 93)(88 94)(89 95)(90 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 51 15 48)(2 52 16 43)(3 53 17 44)(4 54 18 45)(5 49 13 46)(6 50 14 47)(7 57 93 66)(8 58 94 61)(9 59 95 62)(10 60 96 63)(11 55 91 64)(12 56 92 65)(19 40 28 31)(20 41 29 32)(21 42 30 33)(22 37 25 34)(23 38 26 35)(24 39 27 36)(67 79 76 88)(68 80 77 89)(69 81 78 90)(70 82 73 85)(71 83 74 86)(72 84 75 87)
(1 75 15 72)(2 76 16 67)(3 77 17 68)(4 78 18 69)(5 73 13 70)(6 74 14 71)(7 42 93 33)(8 37 94 34)(9 38 95 35)(10 39 96 36)(11 40 91 31)(12 41 92 32)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 61 25 58)(23 62 26 59)(24 63 27 60)(43 88 52 79)(44 89 53 80)(45 90 54 81)(46 85 49 82)(47 86 50 83)(48 87 51 84)

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

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

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

Q8×C2×C6 is a maximal subgroup of
(C6×Q8)⋊6C4  (C3×Q8)⋊13D4  (C2×C6)⋊8Q16  (C6×Q8)⋊7C4  C22.52(S3×Q8)  (C22×Q8)⋊9S3  C6.422- 1+4  C6.442- 1+4  C6.452- 1+4  C22⋊(Q8⋊C9)

60 conjugacy classes

 class 1 2A ··· 2G 3A 3B 4A ··· 4L 6A ··· 6N 12A ··· 12X order 1 2 ··· 2 3 3 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 type + + + - image C1 C2 C2 C3 C6 C6 Q8 C3×Q8 kernel Q8×C2×C6 C22×C12 C6×Q8 C22×Q8 C22×C4 C2×Q8 C2×C6 C22 # reps 1 3 12 2 6 24 4 8

Matrix representation of Q8×C2×C6 in GL4(𝔽13) generated by

 1 0 0 0 0 12 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 1 0 0 0 0 10 0 0 0 0 10
,
 12 0 0 0 0 1 0 0 0 0 0 1 0 0 12 0
,
 12 0 0 0 0 1 0 0 0 0 9 10 0 0 10 4
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,1,0,0,0,0,10,0,0,0,0,10],[12,0,0,0,0,1,0,0,0,0,0,12,0,0,1,0],[12,0,0,0,0,1,0,0,0,0,9,10,0,0,10,4] >;

Q8×C2×C6 in GAP, Magma, Sage, TeX

Q_8\times C_2\times C_6
% in TeX

G:=Group("Q8xC2xC6");
// GroupNames label

G:=SmallGroup(96,222);
// by ID

G=gap.SmallGroup(96,222);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-2,288,601,295]);
// Polycyclic

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

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