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## G = C2×C12.23D4order 192 = 26·3

### Direct product of C2 and C12.23D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C12.23D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×D12 — C2×C12.23D4
 Lower central C3 — C2×C6 — C2×C12.23D4
 Upper central C1 — C23 — C22×Q8

Generators and relations for C2×C12.23D4
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=b6c-1 >

Subgroups: 904 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], S3 [×4], C6, C6 [×6], C2×C4 [×10], C2×C4 [×12], D4 [×8], Q8 [×8], C23, C23 [×16], Dic3 [×4], C12 [×4], C12 [×4], D6 [×20], C2×C6, C2×C6 [×6], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×8], C2×Q8 [×4], C2×Q8 [×4], C24 [×2], D12 [×8], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×10], C2×C12 [×4], C3×Q8 [×8], C22×S3 [×4], C22×S3 [×12], C22×C6, C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, C4×Dic3 [×4], D6⋊C4 [×16], C2×D12 [×4], C2×D12 [×4], C22×Dic3 [×2], C22×C12, C22×C12 [×2], C6×Q8 [×4], C6×Q8 [×4], S3×C23 [×2], C2×C4.4D4, C2×C4×Dic3, C2×D6⋊C4 [×4], C12.23D4 [×8], C22×D12, Q8×C2×C6, C2×C12.23D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C3⋊D4 [×4], C22×S3 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], Q83S3 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C4.4D4, C12.23D4 [×4], C2×Q83S3 [×2], C22×C3⋊D4, C2×C12.23D4

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 12 12 12 12 2 2 2 2 2 4 4 4 4 6 ··· 6 2 ··· 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 C3⋊D4 Q8⋊3S3 kernel C2×C12.23D4 C2×C4×Dic3 C2×D6⋊C4 C12.23D4 C22×D12 Q8×C2×C6 C22×Q8 C2×C12 C22×C4 C2×Q8 C2×C6 C2×C4 C22 # reps 1 1 4 8 1 1 1 4 3 4 8 8 4

Matrix representation of C2×C12.23D4 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 12 0 0 0 0 0 0 12
,
 12 1 0 0 0 0 11 1 0 0 0 0 0 0 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 1
,
 8 5 0 0 0 0 3 5 0 0 0 0 0 0 0 5 0 0 0 0 8 0 0 0 0 0 0 0 9 2 0 0 0 0 11 4
,
 1 12 0 0 0 0 0 12 0 0 0 0 0 0 12 0 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))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,11,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[8,3,0,0,0,0,5,5,0,0,0,0,0,0,0,8,0,0,0,0,5,0,0,0,0,0,0,0,9,11,0,0,0,0,2,4],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

C2×C12.23D4 in GAP, Magma, Sage, TeX

C_2\times C_{12}._{23}D_4
% in TeX

G:=Group("C2xC12.23D4");
// GroupNames label

G:=SmallGroup(192,1373);
// by ID

G=gap.SmallGroup(192,1373);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,297,136,6278]);
// Polycyclic

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

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