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## G = D12.35C23order 192 = 26·3

### 16th non-split extension by D12 of C23 acting via C23/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D12.35C23
 Chief series C1 — C3 — C6 — C12 — D12 — C4○D12 — Q8○D12 — D12.35C23
 Lower central C3 — C6 — C12 — D12.35C23
 Upper central C1 — C2 — C4○D4 — 2- 1+4

Generators and relations for D12.35C23
G = < a,b,c,d,e | a12=b2=e2=1, c2=d2=a6, bab=a-1, ac=ca, ad=da, eae=a7, bc=cb, bd=db, ebe=a3b, dcd-1=a6c, ce=ec, de=ed >

Subgroups: 536 in 248 conjugacy classes, 107 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, D4, Q8, Q8, Q8, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×Q8, C2×Q8, C4○D4, C4○D4, C4○D4, C3⋊C8, C3⋊C8, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×D4, C3×Q8, C3×Q8, C3×Q8, C8○D4, C2×Q16, C4○D8, C8.C22, 2- 1+4, 2- 1+4, C2×C3⋊C8, C4.Dic3, D4⋊S3, D4.S3, Q82S3, C3⋊Q16, C2×Dic6, C4○D12, D42S3, S3×Q8, C6×Q8, C6×Q8, C3×C4○D4, C3×C4○D4, C3×C4○D4, Q8○D8, Q8.11D6, C2×C3⋊Q16, D4.Dic3, Q8.13D6, Q8.14D6, Q8○D12, C3×2- 1+4, D12.35C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C3⋊D4, S3×C23, Q8○D8, C22×C3⋊D4, D12.35C23

Smallest permutation representation of D12.35C23
On 96 points
Generators in S96
```(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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 21)(14 20)(15 19)(16 18)(22 24)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 39)(40 48)(41 47)(42 46)(43 45)(49 60)(50 59)(51 58)(52 57)(53 56)(54 55)(61 63)(64 72)(65 71)(66 70)(67 69)(73 78)(74 77)(75 76)(79 84)(80 83)(81 82)(85 87)(88 96)(89 95)(90 94)(91 93)
(1 55 7 49)(2 56 8 50)(3 57 9 51)(4 58 10 52)(5 59 11 53)(6 60 12 54)(13 70 19 64)(14 71 20 65)(15 72 21 66)(16 61 22 67)(17 62 23 68)(18 63 24 69)(25 79 31 73)(26 80 32 74)(27 81 33 75)(28 82 34 76)(29 83 35 77)(30 84 36 78)(37 85 43 91)(38 86 44 92)(39 87 45 93)(40 88 46 94)(41 89 47 95)(42 90 48 96)
(1 28 7 34)(2 29 8 35)(3 30 9 36)(4 31 10 25)(5 32 11 26)(6 33 12 27)(13 46 19 40)(14 47 20 41)(15 48 21 42)(16 37 22 43)(17 38 23 44)(18 39 24 45)(49 82 55 76)(50 83 56 77)(51 84 57 78)(52 73 58 79)(53 74 59 80)(54 75 60 81)(61 91 67 85)(62 92 68 86)(63 93 69 87)(64 94 70 88)(65 95 71 89)(66 96 72 90)
(1 19)(2 14)(3 21)(4 16)(5 23)(6 18)(7 13)(8 20)(9 15)(10 22)(11 17)(12 24)(25 43)(26 38)(27 45)(28 40)(29 47)(30 42)(31 37)(32 44)(33 39)(34 46)(35 41)(36 48)(49 70)(50 65)(51 72)(52 67)(53 62)(54 69)(55 64)(56 71)(57 66)(58 61)(59 68)(60 63)(73 85)(74 92)(75 87)(76 94)(77 89)(78 96)(79 91)(80 86)(81 93)(82 88)(83 95)(84 90)```

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

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

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

39 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B ··· 6F 8A 8B 8C 8D 8E 12A ··· 12J order 1 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 ··· 6 8 8 8 8 8 12 ··· 12 size 1 1 2 2 2 4 12 2 2 2 2 2 4 4 4 12 12 12 2 4 ··· 4 6 6 12 12 12 4 ··· 4

39 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C3⋊D4 C3⋊D4 Q8○D8 D12.35C23 kernel D12.35C23 Q8.11D6 C2×C3⋊Q16 D4.Dic3 Q8.13D6 Q8.14D6 Q8○D12 C3×2- 1+4 2- 1+4 C3×D4 C3×Q8 C2×Q8 C4○D4 D4 Q8 C3 C1 # reps 1 3 3 1 3 3 1 1 1 3 1 3 4 6 2 2 1

Matrix representation of D12.35C23 in GL6(𝔽73)

 0 72 0 0 0 0 1 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0 0 0 0 0 22 51 0 1 0 0 51 51 72 0
,
 1 72 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 59 51 0 1 0 0 22 14 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 66 64 36 0 0 0 71 4 0 36 0 0 63 54 7 9 0 0 12 70 2 69
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 61 38 67 0 0 0 49 47 0 67 0 0 6 27 12 35 0 0 6 46 24 26
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 57 57 0 0 0 0 57 16 0 0 0 0 5 68 57 57 0 0 68 68 57 16

`G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,72,22,51,0,0,1,0,51,51,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,72,72,0,0,0,0,0,0,0,1,59,22,0,0,1,0,51,14,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,66,71,63,12,0,0,64,4,54,70,0,0,36,0,7,2,0,0,0,36,9,69],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,61,49,6,6,0,0,38,47,27,46,0,0,67,0,12,24,0,0,0,67,35,26],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,57,57,5,68,0,0,57,16,68,68,0,0,0,0,57,57,0,0,0,0,57,16] >;`

D12.35C23 in GAP, Magma, Sage, TeX

`D_{12}._{35}C_2^3`
`% in TeX`

`G:=Group("D12.35C2^3");`
`// GroupNames label`

`G:=SmallGroup(192,1397);`
`// by ID`

`G=gap.SmallGroup(192,1397);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,387,184,675,1684,235,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^12=b^2=e^2=1,c^2=d^2=a^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,e*a*e=a^7,b*c=c*b,b*d=d*b,e*b*e=a^3*b,d*c*d^-1=a^6*c,c*e=e*c,d*e=e*d>;`
`// generators/relations`

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