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## G = (C2×C4).21D12order 192 = 26·3

### 14th non-split extension by C2×C4 of D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C4).21D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C2×D6⋊C4 — (C2×C4).21D12
 Lower central C3 — C22×C6 — (C2×C4).21D12
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C4).21D12
G = < a,b,c,d | a2=b4=c12=1, d2=ab2, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=ab2c-1 >

Subgroups: 528 in 170 conjugacy classes, 57 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C4⋊Dic3, D6⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C23.11D4, C6.C42, C3×C2.C42, C2×C4⋊Dic3, C2×D6⋊C4, C2×D6⋊C4, (C2×C4).21D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C22.D4, C4.4D4, C422C2, C2×D12, C4○D12, S3×D4, D42S3, Q83S3, C23.11D4, C427S3, C23.9D6, C23.21D6, C12⋊D4, C4⋊C4⋊S3, (C2×C4).21D12

Smallest permutation representation of (C2×C4).21D12
On 96 points
Generators in S96
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(25 45)(26 46)(27 47)(28 48)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)(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 40 15 58)(2 33 16 61)(3 42 17 60)(4 35 18 63)(5 44 19 50)(6 25 20 65)(7 46 21 52)(8 27 22 67)(9 48 23 54)(10 29 24 69)(11 38 13 56)(12 31 14 71)(26 75 66 91)(28 77 68 93)(30 79 70 95)(32 81 72 85)(34 83 62 87)(36 73 64 89)(37 78 55 94)(39 80 57 96)(41 82 59 86)(43 84 49 88)(45 74 51 90)(47 76 53 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 80 81 12)(2 11 82 79)(3 78 83 10)(4 9 84 77)(5 76 73 8)(6 7 74 75)(13 86 95 16)(14 15 96 85)(17 94 87 24)(18 23 88 93)(19 92 89 22)(20 21 90 91)(25 52 51 26)(27 50 53 36)(28 35 54 49)(29 60 55 34)(30 33 56 59)(31 58 57 32)(37 62 69 42)(38 41 70 61)(39 72 71 40)(43 68 63 48)(44 47 64 67)(45 66 65 46)

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3 4A ··· 4F 4G ··· 4L 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 12 12 2 4 ··· 4 12 ··· 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - + image C1 C2 C2 C2 C2 S3 D4 D4 D6 C4○D4 D12 C4○D12 S3×D4 D4⋊2S3 Q8⋊3S3 kernel (C2×C4).21D12 C6.C42 C3×C2.C42 C2×C4⋊Dic3 C2×D6⋊C4 C2.C42 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C22 C22 C22 C22 # reps 1 2 1 1 3 1 2 2 3 10 4 8 1 2 1

Matrix representation of (C2×C4).21D12 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 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 8 5 0 0 0 0 0 0 8 12 0 0 0 0 11 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 3 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 2 8 0 0 0 0 0 0 7 3 0 0 0 0 10 10
,
 5 3 0 0 0 0 5 8 0 0 0 0 0 0 5 0 0 0 0 0 0 5 0 0 0 0 0 0 10 10 0 0 0 0 7 3

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,8,0,0,0,0,0,5,0,0,0,0,0,0,8,11,0,0,0,0,12,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,3,8,0,0,0,0,0,0,5,2,0,0,0,0,0,8,0,0,0,0,0,0,7,10,0,0,0,0,3,10],[5,5,0,0,0,0,3,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,10,7,0,0,0,0,10,3] >;

(C2×C4).21D12 in GAP, Magma, Sage, TeX

(C_2\times C_4)._{21}D_{12}
% in TeX

G:=Group("(C2xC4).21D12");
// GroupNames label

G:=SmallGroup(192,233);
// by ID

G=gap.SmallGroup(192,233);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,64,254,387,100,6278]);
// Polycyclic

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

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