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## G = C2×C12.55D4order 192 = 26·3

### Direct product of C2 and C12.55D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C12.55D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×C3⋊C8 — C22×C3⋊C8 — C2×C12.55D4
 Lower central C3 — C6 — C2×C12.55D4
 Upper central C1 — C22×C4 — C23×C4

Generators and relations for C2×C12.55D4
G = < a,b,c,d | a2=b12=1, c4=b6, d2=b9, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b5, dcd-1=b3c3 >

Subgroups: 344 in 202 conjugacy classes, 103 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C22×C4, C22×C4, C22×C4, C24, C3⋊C8, C2×C12, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C22⋊C8, C22×C8, C23×C4, C2×C3⋊C8, C2×C3⋊C8, C22×C12, C22×C12, C22×C12, C23×C6, C2×C22⋊C8, C12.55D4, C22×C3⋊C8, C23×C12, C2×C12.55D4
Quotients:

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

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

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

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

72 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I 4J 4K 4L 6A ··· 6O 8A ··· 8P 12A ··· 12P order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 4 4 4 6 ··· 6 8 ··· 8 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 1 ··· 1 2 2 2 2 2 ··· 2 6 ··· 6 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C4 C4 C8 S3 D4 Dic3 D6 Dic3 M4(2) C3⋊D4 C3⋊C8 C4.Dic3 kernel C2×C12.55D4 C12.55D4 C22×C3⋊C8 C23×C12 C22×C12 C23×C6 C22×C6 C23×C4 C2×C12 C22×C4 C22×C4 C24 C2×C6 C2×C4 C23 C22 # reps 1 4 2 1 6 2 16 1 4 3 3 1 4 8 8 8

Matrix representation of C2×C12.55D4 in GL5(𝔽73)

 72 0 0 0 0 0 72 0 0 0 0 0 72 0 0 0 0 0 1 0 0 0 0 0 1
,
 72 0 0 0 0 0 0 46 0 0 0 27 46 0 0 0 0 0 24 0 0 0 0 0 3
,
 27 0 0 0 0 0 45 27 0 0 0 72 28 0 0 0 0 0 0 72 0 0 0 46 0
,
 46 0 0 0 0 0 45 27 0 0 0 72 28 0 0 0 0 0 0 72 0 0 0 27 0

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,0,27,0,0,0,46,46,0,0,0,0,0,24,0,0,0,0,0,3],[27,0,0,0,0,0,45,72,0,0,0,27,28,0,0,0,0,0,0,46,0,0,0,72,0],[46,0,0,0,0,0,45,72,0,0,0,27,28,0,0,0,0,0,0,27,0,0,0,72,0] >;

C2×C12.55D4 in GAP, Magma, Sage, TeX

C_2\times C_{12}._{55}D_4
% in TeX

G:=Group("C2xC12.55D4");
// GroupNames label

G:=SmallGroup(192,765);
// by ID

G=gap.SmallGroup(192,765);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=1,c^4=b^6,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^5,d*c*d^-1=b^3*c^3>;
// generators/relations

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