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## G = C6.112+ 1+4order 192 = 26·3

### 11st non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.112+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C4⋊C4⋊7S3 — C6.112+ 1+4
 Lower central C3 — C2×C6 — C6.112+ 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C6.112+ 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=a3b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=b2d >

Subgroups: 616 in 238 conjugacy classes, 99 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C4⋊C4, C42⋊C2, C4×D4, C4⋊D4, C22.D4, C42.C2, C422C2, C4×Dic3, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C4⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C22×C12, C22.47C24, C4.Dic6, C4⋊C47S3, Dic35D4, D6.D4, C12⋊D4, C4⋊C4⋊S3, C4×C3⋊D4, C4×C3⋊D4, C127D4, C127D4, C6×C4⋊C4, C6.112+ 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, C4○D12, Q83S3, S3×C23, C22.47C24, C2×C4○D12, D46D6, C2×Q83S3, C6.112+ 1+4

Smallest permutation representation of C6.112+ 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 53 17 48)(2 52 18 47)(3 51 13 46)(4 50 14 45)(5 49 15 44)(6 54 16 43)(7 61 94 56)(8 66 95 55)(9 65 96 60)(10 64 91 59)(11 63 92 58)(12 62 93 57)(19 32 30 37)(20 31 25 42)(21 36 26 41)(22 35 27 40)(23 34 28 39)(24 33 29 38)(67 80 78 85)(68 79 73 90)(69 84 74 89)(70 83 75 88)(71 82 76 87)(72 81 77 86)
(1 24)(2 19)(3 20)(4 21)(5 22)(6 23)(7 87)(8 88)(9 89)(10 90)(11 85)(12 86)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 60 17 65)(2 55 18 66)(3 56 13 61)(4 57 14 62)(5 58 15 63)(6 59 16 64)(7 46 94 51)(8 47 95 52)(9 48 96 53)(10 43 91 54)(11 44 92 49)(12 45 93 50)(19 67 30 78)(20 68 25 73)(21 69 26 74)(22 70 27 75)(23 71 28 76)(24 72 29 77)(31 79 42 90)(32 80 37 85)(33 81 38 86)(34 82 39 87)(35 83 40 88)(36 84 41 89)
(1 86 4 89)(2 87 5 90)(3 88 6 85)(7 22 10 19)(8 23 11 20)(9 24 12 21)(13 83 16 80)(14 84 17 81)(15 79 18 82)(25 95 28 92)(26 96 29 93)(27 91 30 94)(31 66 34 63)(32 61 35 64)(33 62 36 65)(37 56 40 59)(38 57 41 60)(39 58 42 55)(43 78 46 75)(44 73 47 76)(45 74 48 77)(49 68 52 71)(50 69 53 72)(51 70 54 67)

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 3 4A ··· 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 6A ··· 6G 12A ··· 12L order 1 2 2 2 2 2 2 2 2 3 4 ··· 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 12 12 12 2 2 ··· 2 4 4 4 6 6 6 6 12 12 12 2 ··· 2 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 C4○D4 C4○D12 2+ 1+4 Q8⋊3S3 D4⋊6D6 kernel C6.112+ 1+4 C4.Dic6 C4⋊C4⋊7S3 Dic3⋊5D4 D6.D4 C12⋊D4 C4⋊C4⋊S3 C4×C3⋊D4 C12⋊7D4 C6×C4⋊C4 C2×C4⋊C4 C4⋊C4 C22×C4 C12 C2×C6 C4 C6 C22 C2 # reps 1 1 1 1 2 1 2 3 3 1 1 4 3 4 4 8 1 2 2

Matrix representation of C6.112+ 1+4 in GL4(𝔽13) generated by

 0 1 0 0 12 1 0 0 0 0 12 0 0 0 0 12
,
 3 3 0 0 6 10 0 0 0 0 5 0 0 0 0 5
,
 2 9 0 0 4 11 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 1 10 0 0 5 12
,
 5 0 0 0 0 5 0 0 0 0 8 2 0 0 0 5
G:=sub<GL(4,GF(13))| [0,12,0,0,1,1,0,0,0,0,12,0,0,0,0,12],[3,6,0,0,3,10,0,0,0,0,5,0,0,0,0,5],[2,4,0,0,9,11,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,1,5,0,0,10,12],[5,0,0,0,0,5,0,0,0,0,8,0,0,0,2,5] >;

C6.112+ 1+4 in GAP, Magma, Sage, TeX

C_6._{11}2_+^{1+4}
% in TeX

G:=Group("C6.11ES+(2,2)");
// GroupNames label

G:=SmallGroup(192,1073);
// by ID

G=gap.SmallGroup(192,1073);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,100,1571,6278]);
// Polycyclic

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

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