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## G = C6.812- 1+4order 192 = 26·3

### 36th non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C4○D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.812- 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — Dic3.D4 — C6.812- 1+4
 Lower central C3 — C2×C6 — C6.812- 1+4
 Upper central C1 — C22 — C22.D4

Generators and relations for C6.812- 1+4
G = < a,b,c,d,e | a6=b4=1, c2=a3, d2=e2=a3b2, bab-1=cac-1=eae-1=a-1, ad=da, cbc-1=b-1, bd=db, ebe-1=a3b, dcd-1=a3c, ce=ec, ede-1=a3b2d >

Subgroups: 448 in 196 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C22×C6, C22⋊Q8, C22.D4, C22.D4, C4.4D4, C42.C2, C422C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×D4, C22.57C24, Dic3.D4, C23.8D6, C12⋊Q8, Dic3.Q8, C12.48D4, C23.23D6, C23.12D6, C3×C22.D4, C6.812- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, 2- 1+4, S3×C23, C22.57C24, D46D6, Q8○D12, C6.812- 1+4

Smallest permutation representation of C6.812- 1+4
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 85 14 84)(2 90 15 83)(3 89 16 82)(4 88 17 81)(5 87 18 80)(6 86 13 79)(7 24 93 26)(8 23 94 25)(9 22 95 30)(10 21 96 29)(11 20 91 28)(12 19 92 27)(31 59 39 66)(32 58 40 65)(33 57 41 64)(34 56 42 63)(35 55 37 62)(36 60 38 61)(43 75 52 68)(44 74 53 67)(45 73 54 72)(46 78 49 71)(47 77 50 70)(48 76 51 69)
(1 72 4 69)(2 71 5 68)(3 70 6 67)(7 38 10 41)(8 37 11 40)(9 42 12 39)(13 74 16 77)(14 73 17 76)(15 78 18 75)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 62 28 65)(26 61 29 64)(27 66 30 63)(31 95 34 92)(32 94 35 91)(33 93 36 96)(43 83 46 80)(44 82 47 79)(45 81 48 84)(49 87 52 90)(50 86 53 89)(51 85 54 88)
(1 26 17 21)(2 27 18 22)(3 28 13 23)(4 29 14 24)(5 30 15 19)(6 25 16 20)(7 81 96 85)(8 82 91 86)(9 83 92 87)(10 84 93 88)(11 79 94 89)(12 80 95 90)(31 49 42 43)(32 50 37 44)(33 51 38 45)(34 52 39 46)(35 53 40 47)(36 54 41 48)(55 67 65 77)(56 68 66 78)(57 69 61 73)(58 70 62 74)(59 71 63 75)(60 72 64 76)
(1 93 17 10)(2 92 18 9)(3 91 13 8)(4 96 14 7)(5 95 15 12)(6 94 16 11)(19 90 30 80)(20 89 25 79)(21 88 26 84)(22 87 27 83)(23 86 28 82)(24 85 29 81)(31 75 42 71)(32 74 37 70)(33 73 38 69)(34 78 39 68)(35 77 40 67)(36 76 41 72)(43 59 49 63)(44 58 50 62)(45 57 51 61)(46 56 52 66)(47 55 53 65)(48 60 54 64)

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 D6 2+ 1+4 2- 1+4 D4⋊6D6 Q8○D12 kernel C6.812- 1+4 Dic3.D4 C23.8D6 C12⋊Q8 Dic3.Q8 C12.48D4 C23.23D6 C23.12D6 C3×C22.D4 C22.D4 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C6 C6 C2 C2 # reps 1 2 4 2 2 2 1 1 1 1 3 2 1 1 1 2 2 4

Matrix representation of C6.812- 1+4 in GL8(𝔽13)

 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12
,
 0 0 1 12 0 0 0 0 0 0 0 12 0 0 0 0 12 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0
,
 2 2 0 0 0 0 0 0 4 11 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 4 11 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0
,
 2 9 0 0 0 0 0 0 4 11 0 0 0 0 0 0 0 0 11 4 0 0 0 0 0 0 9 2 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0
,
 0 0 2 2 0 0 0 0 0 0 4 11 0 0 0 0 11 11 0 0 0 0 0 0 9 2 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 5

G:=sub<GL(8,GF(13))| [0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0],[2,4,0,0,0,0,0,0,2,11,0,0,0,0,0,0,0,0,2,4,0,0,0,0,0,0,2,11,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0],[2,4,0,0,0,0,0,0,9,11,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[0,0,11,9,0,0,0,0,0,0,11,2,0,0,0,0,2,4,0,0,0,0,0,0,2,11,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,5] >;

C6.812- 1+4 in GAP, Magma, Sage, TeX

C_6._{81}2_-^{1+4}
% in TeX

G:=Group("C6.81ES-(2,2)");
// GroupNames label

G:=SmallGroup(192,1210);
// by ID

G=gap.SmallGroup(192,1210);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,219,184,1571,570,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^4=1,c^2=a^3,d^2=e^2=a^3*b^2,b*a*b^-1=c*a*c^-1=e*a*e^-1=a^-1,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e^-1=a^3*b,d*c*d^-1=a^3*c,c*e=e*c,e*d*e^-1=a^3*b^2*d>;
// generators/relations

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