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## G = (C2×C12)⋊17D4order 192 = 26·3

### 13rd semidirect product of C2×C12 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×C12)⋊17D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — (C2×C12)⋊17D4
 Lower central C3 — C2×C6 — (C2×C12)⋊17D4
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for (C2×C12)⋊17D4
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, dad=ab6, cbc-1=dbd=b5, dcd=c-1 >

Subgroups: 744 in 310 conjugacy classes, 115 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×C6, C22×C6, C2×C42, C4×D4, C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×C4○D4, C2×C4○D4, C4×Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C22×Dic3, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C22.26C24, C2×C4×Dic3, C4×C3⋊D4, C23.12D6, C23.14D6, C123D4, Dic3⋊Q8, C12.23D4, C2×C4○D12, C6×C4○D4, (C2×C12)⋊17D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C4○D4, C2×C3⋊D4, S3×C23, C22.26C24, S3×C4○D4, C22×C3⋊D4, (C2×C12)⋊17D4

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 S3×C4○D4 kernel (C2×C12)⋊17D4 C2×C4×Dic3 C4×C3⋊D4 C23.12D6 C23.14D6 C12⋊3D4 Dic3⋊Q8 C12.23D4 C2×C4○D12 C6×C4○D4 C2×C4○D4 C2×C12 C22×C4 C2×D4 C2×Q8 Dic3 C2×C4 C2 # reps 1 1 4 1 4 1 1 1 1 1 1 4 3 3 1 8 8 4

Matrix representation of (C2×C12)⋊17D4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 3 0 0 0 0 5 5
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 1 0 0 0 0 0 0 5 0 0 0 0 0 0 5
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 2 0 0 0 0 0 12

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,5,0,0,0,0,3,5],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,2,12] >;

(C2×C12)⋊17D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})\rtimes_{17}D_4
% in TeX

G:=Group("(C2xC12):17D4");
// GroupNames label

G:=SmallGroup(192,1391);
// by ID

G=gap.SmallGroup(192,1391);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,297,6278]);
// Polycyclic

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

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