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## G = (C22×C4).37D6order 192 = 26·3

### 21st non-split extension by C22×C4 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C22×C4).37D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C2×D6⋊C4 — (C22×C4).37D6
 Lower central C3 — C22×C6 — (C22×C4).37D6
 Upper central C1 — C23 — C2.C42

Generators and relations for (C22×C4).37D6
G = < a,b,c,d,e | a2=b2=c4=1, d6=a, e2=abc2, ab=ba, ac=ca, ad=da, ae=ea, dcd-1=bc=cb, bd=db, be=eb, ece-1=abc-1, ede-1=bc2d5 >

Subgroups: 560 in 186 conjugacy classes, 59 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, Dic3⋊C4, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C23.Q8, C3×C2.C42, C2×Dic3⋊C4, C2×D6⋊C4, (C22×C4).37D6
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C22×S3, C4⋊D4, C22⋊Q8, C422C2, C4○D12, S3×D4, S3×Q8, C23.Q8, C423S3, Dic3⋊D4, D6⋊Q8, (C22×C4).37D6

Smallest permutation representation of (C22×C4).37D6
On 96 points
Generators in S96
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 73)(9 74)(10 75)(11 76)(12 77)(13 44)(14 45)(15 46)(16 47)(17 48)(18 37)(19 38)(20 39)(21 40)(22 41)(23 42)(24 43)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(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 69 57 20)(2 92 58 40)(3 71 59 22)(4 94 60 42)(5 61 49 24)(6 96 50 44)(7 63 51 14)(8 86 52 46)(9 65 53 16)(10 88 54 48)(11 67 55 18)(12 90 56 38)(13 83 62 26)(15 73 64 28)(17 75 66 30)(19 77 68 32)(21 79 70 34)(23 81 72 36)(25 43 82 95)(27 45 84 85)(29 47 74 87)(31 37 76 89)(33 39 78 91)(35 41 80 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 6 27 32)(2 31 28 5)(3 4 29 30)(7 12 33 26)(8 25 34 11)(9 10 35 36)(13 39 90 63)(14 62 91 38)(15 37 92 61)(16 72 93 48)(17 47 94 71)(18 70 95 46)(19 45 96 69)(20 68 85 44)(21 43 86 67)(22 66 87 42)(23 41 88 65)(24 64 89 40)(49 58 76 73)(50 84 77 57)(51 56 78 83)(52 82 79 55)(53 54 80 81)(59 60 74 75)

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

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

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

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 2 2 2 2 2 2 4 4 type + + + + + + - + + - image C1 C2 C2 C2 S3 D4 Q8 D6 C4○D4 C4○D12 S3×D4 S3×Q8 kernel (C22×C4).37D6 C3×C2.C42 C2×Dic3⋊C4 C2×D6⋊C4 C2.C42 C2×Dic3 C22×S3 C22×C4 C2×C6 C22 C22 C22 # reps 1 1 3 3 1 6 2 3 6 12 3 1

Matrix representation of (C22×C4).37D6 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 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 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 12 0 0 0 0 2 1 0 0 0 0 0 0 2 4 0 0 0 0 9 11
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 5 5 0 0 0 0 3 8 0 0 0 0 0 0 8 8 0 0 0 0 5 0
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 8 8 0 0 0 0 0 5 0 0 0 0 0 0 8 8 0 0 0 0 0 5

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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,2,0,0,0,0,12,1,0,0,0,0,0,0,2,9,0,0,0,0,4,11],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,5,3,0,0,0,0,5,8,0,0,0,0,0,0,8,5,0,0,0,0,8,0],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,8,5,0,0,0,0,0,0,8,0,0,0,0,0,8,5] >;

(C22×C4).37D6 in GAP, Magma, Sage, TeX

(C_2^2\times C_4)._{37}D_6
% in TeX

G:=Group("(C2^2xC4).37D6");
// GroupNames label

G:=SmallGroup(192,235);
// by ID

G=gap.SmallGroup(192,235);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,64,1262,387,268,6278]);
// Polycyclic

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

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