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## G = C3×C4⋊Q8order 96 = 25·3

### Direct product of C3 and C4⋊Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C3×C4⋊Q8
 Chief series C1 — C2 — C22 — C2×C6 — C2×C12 — C6×Q8 — C3×C4⋊Q8
 Lower central C1 — C22 — C3×C4⋊Q8
 Upper central C1 — C2×C6 — C3×C4⋊Q8

Generators and relations for C3×C4⋊Q8
G = < a,b,c,d | a3=b4=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 84 in 68 conjugacy classes, 52 normal (12 characteristic)
C1, C2, C2 [×2], C3, C4 [×6], C4 [×4], C22, C6, C6 [×2], C2×C4, C2×C4 [×6], Q8 [×4], C12 [×6], C12 [×4], C2×C6, C42, C4⋊C4 [×4], C2×Q8 [×2], C2×C12, C2×C12 [×6], C3×Q8 [×4], C4⋊Q8, C4×C12, C3×C4⋊C4 [×4], C6×Q8 [×2], C3×C4⋊Q8
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], D4 [×2], Q8 [×4], C23, C2×C6 [×7], C2×D4, C2×Q8 [×2], C3×D4 [×2], C3×Q8 [×4], C22×C6, C4⋊Q8, C6×D4, C6×Q8 [×2], C3×C4⋊Q8

Smallest permutation representation of C3×C4⋊Q8
Regular action on 96 points
Generators in S96
(1 26 14)(2 27 15)(3 28 16)(4 25 13)(5 34 42)(6 35 43)(7 36 44)(8 33 41)(9 63 46)(10 64 47)(11 61 48)(12 62 45)(17 22 30)(18 23 31)(19 24 32)(20 21 29)(37 94 60)(38 95 57)(39 96 58)(40 93 59)(49 84 70)(50 81 71)(51 82 72)(52 83 69)(53 87 91)(54 88 92)(55 85 89)(56 86 90)(65 76 80)(66 73 77)(67 74 78)(68 75 79)
(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 32 12 34)(2 29 9 35)(3 30 10 36)(4 31 11 33)(5 14 24 45)(6 15 21 46)(7 16 22 47)(8 13 23 48)(17 64 44 28)(18 61 41 25)(19 62 42 26)(20 63 43 27)(37 82 77 89)(38 83 78 90)(39 84 79 91)(40 81 80 92)(49 75 87 58)(50 76 88 59)(51 73 85 60)(52 74 86 57)(53 96 70 68)(54 93 71 65)(55 94 72 66)(56 95 69 67)
(1 57 12 74)(2 60 9 73)(3 59 10 76)(4 58 11 75)(5 69 24 56)(6 72 21 55)(7 71 22 54)(8 70 23 53)(13 96 48 68)(14 95 45 67)(15 94 46 66)(16 93 47 65)(17 92 44 81)(18 91 41 84)(19 90 42 83)(20 89 43 82)(25 39 61 79)(26 38 62 78)(27 37 63 77)(28 40 64 80)(29 85 35 51)(30 88 36 50)(31 87 33 49)(32 86 34 52)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A ··· 4F 4G 4H 4I 4J 6A ··· 6F 12A ··· 12L 12M ··· 12T order 1 2 2 2 3 3 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 1 1 1 1 2 ··· 2 4 4 4 4 1 ··· 1 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C3 C6 C6 C6 D4 Q8 C3×D4 C3×Q8 kernel C3×C4⋊Q8 C4×C12 C3×C4⋊C4 C6×Q8 C4⋊Q8 C42 C4⋊C4 C2×Q8 C12 C12 C4 C4 # reps 1 1 4 2 2 2 8 4 2 4 4 8

Matrix representation of C3×C4⋊Q8 in GL5(𝔽13)

 9 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 2 0 0 0 12 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 1 0
,
 1 0 0 0 0 0 12 2 0 0 0 0 1 0 0 0 0 0 0 5 0 0 0 5 0

G:=sub<GL(5,GF(13))| [9,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,12,0,0,0,2,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,12,0],[1,0,0,0,0,0,12,0,0,0,0,2,1,0,0,0,0,0,0,5,0,0,0,5,0] >;

C3×C4⋊Q8 in GAP, Magma, Sage, TeX

C_3\times C_4\rtimes Q_8
% in TeX

G:=Group("C3xC4:Q8");
// GroupNames label

G:=SmallGroup(96,175);
// by ID

G=gap.SmallGroup(96,175);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,144,313,151,938,230]);
// Polycyclic

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

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