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## G = C22×C4○D12order 192 = 26·3

### Direct product of C22 and C4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C22×C4○D12
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — S3×C22×C4 — C22×C4○D12
 Lower central C3 — C6 — C22×C4○D12
 Upper central C1 — C22×C4 — C23×C4

Generators and relations for C22×C4○D12
G = < a,b,c,d,e | a2=b2=c4=e2=1, d6=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d5 >

Subgroups: 1784 in 890 conjugacy classes, 463 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C23×C4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C22×Dic3, C2×C3⋊D4, C22×C12, C22×C12, S3×C23, C23×C6, C22×C4○D4, C22×Dic6, S3×C22×C4, C22×D12, C2×C4○D12, C22×C3⋊D4, C23×C12, C22×C4○D12
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, C25, C4○D12, S3×C23, C22×C4○D4, C2×C4○D12, S3×C24, C22×C4○D12

Smallest permutation representation of C22×C4○D12
On 96 points
Generators in S96
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 32)(14 33)(15 34)(16 35)(17 36)(18 25)(19 26)(20 27)(21 28)(22 29)(23 30)(24 31)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 95)(62 96)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)
(1 72)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 96)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)
(1 74 7 80)(2 75 8 81)(3 76 9 82)(4 77 10 83)(5 78 11 84)(6 79 12 73)(13 94 19 88)(14 95 20 89)(15 96 21 90)(16 85 22 91)(17 86 23 92)(18 87 24 93)(25 65 31 71)(26 66 32 72)(27 67 33 61)(28 68 34 62)(29 69 35 63)(30 70 36 64)(37 58 43 52)(38 59 44 53)(39 60 45 54)(40 49 46 55)(41 50 47 56)(42 51 48 57)
(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 93)(2 92)(3 91)(4 90)(5 89)(6 88)(7 87)(8 86)(9 85)(10 96)(11 95)(12 94)(13 79)(14 78)(15 77)(16 76)(17 75)(18 74)(19 73)(20 84)(21 83)(22 82)(23 81)(24 80)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 50)(32 49)(33 60)(34 59)(35 58)(36 57)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 72)(47 71)(48 70)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L ··· 2S 3 4A ··· 4H 4I 4J 4K 4L 4M ··· 4T 6A ··· 6O 12A ··· 12P order 1 2 ··· 2 2 2 2 2 2 ··· 2 3 4 ··· 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 6 ··· 6 2 1 ··· 1 2 2 2 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 C4○D12 kernel C22×C4○D12 C22×Dic6 S3×C22×C4 C22×D12 C2×C4○D12 C22×C3⋊D4 C23×C12 C23×C4 C22×C4 C24 C2×C6 C22 # reps 1 1 2 1 24 2 1 1 14 1 8 16

Matrix representation of C22×C4○D12 in GL5(𝔽13)

 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 5 0 0 0 0 0 5
,
 1 0 0 0 0 0 0 1 0 0 0 12 12 0 0 0 0 0 0 5 0 0 0 5 0
,
 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 5 0 0 0 8 0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,5],[1,0,0,0,0,0,0,12,0,0,0,1,12,0,0,0,0,0,0,5,0,0,0,5,0],[1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,5,0] >;

C22×C4○D12 in GAP, Magma, Sage, TeX

C_2^2\times C_4\circ D_{12}
% in TeX

G:=Group("C2^2xC4oD12");
// GroupNames label

G:=SmallGroup(192,1513);
// by ID

G=gap.SmallGroup(192,1513);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,136,1684,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=e^2=1,d^6=c^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d^5>;
// generators/relations

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