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## G = (C2×C12)⋊5D4order 192 = 26·3

### 1st semidirect product of C2×C12 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C12)⋊5D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×D12 — (C2×C12)⋊5D4
 Lower central C3 — C22×C6 — (C2×C12)⋊5D4
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C12)⋊5D4
G = < a,b,c,d | a2=b12=c4=d2=1, cbc-1=ab=ba, ac=ca, ad=da, dbd=b-1, dcd=c-1 >

Subgroups: 1168 in 322 conjugacy classes, 69 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×6], C3, C4 [×7], C22, C22 [×6], C22 [×30], S3 [×6], C6, C6 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×24], C23, C23 [×24], Dic3, C12 [×6], D6 [×30], C2×C6, C2×C6 [×6], C22⋊C4 [×6], C22×C4 [×3], C22×C4, C2×D4 [×18], C24 [×3], D12 [×24], C2×Dic3 [×3], C2×C12 [×6], C2×C12 [×6], C22×S3 [×6], C22×S3 [×18], C22×C6, C2.C42, C2×C22⋊C4 [×3], C22×D4 [×3], D6⋊C4 [×6], C2×D12 [×18], C22×Dic3, C22×C12 [×3], S3×C23 [×3], C232D4, C3×C2.C42, C2×D6⋊C4 [×3], C22×D12 [×3], (C2×C12)⋊5D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×12], C23, D6 [×3], C2×D4 [×6], C4○D4, D12 [×6], C22×S3, C22≀C2 [×3], C4⋊D4 [×3], C41D4, C2×D12 [×3], S3×D4 [×3], Q83S3, C232D4, C4⋊D12, D6⋊D4 [×3], C12⋊D4 [×3], (C2×C12)⋊5D4

Smallest permutation representation of (C2×C12)⋊5D4
On 96 points
Generators in S96
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 60)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 85)(46 86)(47 87)(48 88)(61 81)(62 82)(63 83)(64 84)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 80)
(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 44 50 82)(2 85 51 63)(3 46 52 84)(4 87 53 65)(5 48 54 74)(6 89 55 67)(7 38 56 76)(8 91 57 69)(9 40 58 78)(10 93 59 71)(11 42 60 80)(12 95 49 61)(13 96 27 62)(14 45 28 83)(15 86 29 64)(16 47 30 73)(17 88 31 66)(18 37 32 75)(19 90 33 68)(20 39 34 77)(21 92 35 70)(22 41 36 79)(23 94 25 72)(24 43 26 81)
(1 82)(2 81)(3 80)(4 79)(5 78)(6 77)(7 76)(8 75)(9 74)(10 73)(11 84)(12 83)(13 62)(14 61)(15 72)(16 71)(17 70)(18 69)(19 68)(20 67)(21 66)(22 65)(23 64)(24 63)(25 86)(26 85)(27 96)(28 95)(29 94)(30 93)(31 92)(32 91)(33 90)(34 89)(35 88)(36 87)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 60)(47 59)(48 58)

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2M 3 4A ··· 4F 4G 4H 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 12 ··· 12 2 4 ··· 4 12 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D4 D6 C4○D4 D12 S3×D4 Q8⋊3S3 kernel (C2×C12)⋊5D4 C3×C2.C42 C2×D6⋊C4 C22×D12 C2.C42 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C22 C22 # reps 1 1 3 3 1 6 6 3 2 12 3 1

Matrix representation of (C2×C12)⋊5D4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 2 0 0 0 0 0 12 0 0 0 0 0 0 1 2 0 0 0 0 12 12 0 0 0 0 0 0 3 3 0 0 0 0 10 6
,
 12 11 0 0 0 0 1 1 0 0 0 0 0 0 12 11 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 11 0 0 0 0 0 1 0 0 0 0 0 0 12 11 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,2,12,0,0,0,0,0,0,1,12,0,0,0,0,2,12,0,0,0,0,0,0,3,10,0,0,0,0,3,6],[12,1,0,0,0,0,11,1,0,0,0,0,0,0,12,1,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

(C2×C12)⋊5D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})\rtimes_5D_4
% in TeX

G:=Group("(C2xC12):5D4");
// GroupNames label

G:=SmallGroup(192,230);
// by ID

G=gap.SmallGroup(192,230);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,387,58,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=d^2=1,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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