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G = C2×C4⋊D12  order 192 = 26·3

Direct product of C2 and C4⋊D12

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C4⋊D12, C42⋊41D6, C4⋊2(C2×D12), (C2×C4)⋊7D12, C12⋊10(C2×D4), (C2×C12)⋊30D4, C6⋊1(C4⋊1D4), (C2×C42)⋊10S3, C6.3(C22×D4), (C4×C12)⋊50C22, (C22×D12)⋊4C2, (C2×C6).19C24, C2.5(C22×D12), (C2×D12)⋊42C22, (C22×C4).454D6, C22.64(C2×D12), (C2×C12).780C23, (C22×S3).1C23, C22.62(S3×C23), (S3×C23).27C22, (C22×C6).381C23, C23.325(C22×S3), (C22×C12).523C22, (C2×C4×C12)⋊8C2, C3⋊1(C2×C4⋊1D4), (C2×C6).170(C2×D4), (C2×C4).729(C22×S3), SmallGroup(192,1034)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C2×C4⋊D12
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×D12 — C2×C4⋊D12
Lower central
C3 — C2×C6 — C2×C4⋊D12

Subgroups: 1656 in 498 conjugacy classes, 159 normal (9 characteristic)
C1, C2 [×7], C2 [×8], C3, C4 [×12], C22, C22 [×6], C22 [×40], S3 [×8], C6 [×7], C2×C4 [×18], D4 [×48], C23, C23 [×32], C12 [×12], D6 [×40], C2×C6, C2×C6 [×6], C42 [×4], C22×C4 [×3], C2×D4 [×48], C24 [×4], D12 [×48], C2×C12 [×18], C22×S3 [×8], C22×S3 [×24], C22×C6, C2×C42, C4⋊1D4 [×8], C22×D4 [×6], C4×C12 [×4], C2×D12 [×24], C2×D12 [×24], C22×C12 [×3], S3×C23 [×4], C2×C4⋊1D4, C4⋊D12 [×8], C2×C4×C12, C22×D12 [×6], C2×C4⋊D12

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×12], C23 [×15], D6 [×7], C2×D4 [×18], C24, D12 [×12], C22×S3 [×7], C4⋊1D4 [×4], C22×D4 [×3], C2×D12 [×18], S3×C23, C2×C4⋊1D4, C4⋊D12 [×4], C22×D12 [×3], C2×C4⋊D12

Generators and relations
 G = < a,b,c,d | a2=b4=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

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

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

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

Matrix representation ►G ⊆ GL5(𝔽13)

120000
012000
001200
000120
000012
,
10000
00100
012000
00010
00001
,
10000
001200
01000
0001010
00037
,
10000
01000
001200
00037
0001010

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,10,3,0,0,0,10,7],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,3,10,0,0,0,7,10] >;
 

60 conjugacy classes

class 1 2A···2G2H···2O 3 4A···4L6A···6G12A···12X
order12···22···234···46···612···12
size11···112···1222···22···22···2

60 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2S3D4D6D6D12
kernelC2×C4⋊D12C4⋊D12C2×C4×C12C22×D12C2×C42C2×C12C42C22×C4C2×C4
# reps18161124324

In GAP, Magma, Sage, TeX 

C_2\times C_4\rtimes D_{12}
 
% in TeX
 
G:=Group("C2xC4:D12");
 
// GroupNames label
 
G:=SmallGroup(192,1034);
 
// by ID
 
G=gap.SmallGroup(192,1034);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,80,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^4=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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