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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

Aliases: C3×C4⋊Q8, C124Q8, C12.40D4, C42.5C6, C4⋊(C3×Q8), C4⋊C4.5C6, C4.5(C3×D4), C2.5(C6×Q8), C6.73(C2×D4), C2.10(C6×D4), (C6×Q8).8C2, (C2×Q8).5C6, C6.22(C2×Q8), (C4×C12).11C2, (C2×C6).83C23, (C2×C12).126C22, C22.18(C22×C6), (C2×C4).9(C2×C6), (C3×C4⋊C4).12C2, SmallGroup(96,175)

Series: Derived Chief Lower central Upper central

C1C22 — C3×C4⋊Q8
C1C2C22C2×C6C2×C12C6×Q8 — C3×C4⋊Q8
C1C22 — C3×C4⋊Q8
C1C2×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)])

C3×C4⋊Q8 is a maximal subgroup of
C12.5Q16  C12.10D8  C42.3Dic3  C12.17D8  C12.9Q16  C12.SD16  C42.76D6  C42.77D6  C125SD16  D125Q8  C126SD16  C42.80D6  D126Q8  C12.D8  C42.82D6  C12⋊Q16  Dic65Q8  C123Q16  C12.Q16  Dic66Q8  D12.15D4  Dic68Q8  Dic69Q8  C42.171D6  C42.240D6  D1212D4  D128Q8  C42.241D6  C42.174D6  D129Q8  C42.176D6  C42.177D6  C42.178D6  C42.179D6  C42.180D6  C3×D4×Q8  C3×Q82

42 conjugacy classes

class 1 2A2B2C3A3B4A···4F4G4H4I4J6A···6F12A···12L12M···12T
order1222334···444446···612···1212···12
size1111112···244441···12···24···4

42 irreducible representations

dim111111112222
type+++++-
imageC1C2C2C2C3C6C6C6D4Q8C3×D4C3×Q8
kernelC3×C4⋊Q8C4×C12C3×C4⋊C4C6×Q8C4⋊Q8C42C4⋊C4C2×Q8C12C12C4C4
# reps114222842448

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

90000
01000
00100
00010
00001
,
10000
012200
012100
000120
000012
,
10000
01000
00100
000012
00010
,
10000
012200
00100
00005
00050

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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