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## G = C12.88(C2×Q8)  order 192 = 26·3

### 20th non-split extension by C12 of C2×Q8 acting via C2×Q8/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C12.88(C2×Q8)
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4×Dic3 — C23.26D6 — C12.88(C2×Q8)
 Lower central C3 — C2×C6 — C12.88(C2×Q8)
 Upper central C1 — C2×C4 — C2×M4(2)

Generators and relations for C12.88(C2×Q8)
G = < a,b,c,d | a12=c4=1, b2=a6, d2=a9c2, ab=ba, cac-1=a5, ad=da, cbc-1=dbd-1=a6b, dcd-1=a6c-1 >

Subgroups: 216 in 114 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C4⋊C8, C42⋊C2, C22×C8, C2×M4(2), C2×C3⋊C8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C22×C12, C42.6C22, Dic3⋊C8, C22×C3⋊C8, C23.26D6, C6×M4(2), C12.88(C2×Q8)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, C4×S3, C3⋊D4, C22×S3, C2×C4⋊C4, C8○D4, Dic3⋊C4, C2×Dic6, S3×C2×C4, C2×C3⋊D4, C42.6C22, D12.C4, C2×Dic3⋊C4, C12.88(C2×Q8)

Smallest permutation representation of C12.88(C2×Q8)
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 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 48)(49 58 55 52)(50 59 56 53)(51 60 57 54)(61 70 67 64)(62 71 68 65)(63 72 69 66)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 88 91 94)(86 89 92 95)(87 90 93 96)
(1 42 57 78)(2 47 58 83)(3 40 59 76)(4 45 60 81)(5 38 49 74)(6 43 50 79)(7 48 51 84)(8 41 52 77)(9 46 53 82)(10 39 54 75)(11 44 55 80)(12 37 56 73)(13 87 68 25)(14 92 69 30)(15 85 70 35)(16 90 71 28)(17 95 72 33)(18 88 61 26)(19 93 62 31)(20 86 63 36)(21 91 64 29)(22 96 65 34)(23 89 66 27)(24 94 67 32)
(1 30 54 89 7 36 60 95)(2 31 55 90 8 25 49 96)(3 32 56 91 9 26 50 85)(4 33 57 92 10 27 51 86)(5 34 58 93 11 28 52 87)(6 35 59 94 12 29 53 88)(13 44 65 77 19 38 71 83)(14 45 66 78 20 39 72 84)(15 46 67 79 21 40 61 73)(16 47 68 80 22 41 62 74)(17 48 69 81 23 42 63 75)(18 37 70 82 24 43 64 76)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E 8A 8B 8C 8D 8E ··· 8L 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 8 ··· 8 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 2 1 1 1 1 2 2 12 12 12 12 2 2 2 4 4 4 4 4 4 6 ··· 6 2 2 2 2 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 type + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 C4 S3 D4 Q8 D6 D6 Dic6 C4×S3 C3⋊D4 C4×S3 C8○D4 D12.C4 kernel C12.88(C2×Q8) Dic3⋊C8 C22×C3⋊C8 C23.26D6 C6×M4(2) C4⋊Dic3 C6.D4 C2×M4(2) C2×C12 C2×C12 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 4 1 1 1 4 4 1 2 2 2 1 4 2 4 2 8 4

Matrix representation of C12.88(C2×Q8) in GL6(𝔽73)

 0 1 0 0 0 0 72 72 0 0 0 0 0 0 27 0 0 0 0 0 0 27 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 27 0 0 0 0 0 55 46 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 63 22 0 0 0 0 32 10 0 0 0 0 0 0 22 66 0 0 0 0 38 51 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 3 0 0 0 0 15 72 0 0 0 0 0 0 0 72 0 0 0 0 72 0

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,55,0,0,0,0,0,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[63,32,0,0,0,0,22,10,0,0,0,0,0,0,22,38,0,0,0,0,66,51,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,15,0,0,0,0,3,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

C12.88(C2×Q8) in GAP, Magma, Sage, TeX

C_{12}._{88}(C_2\times Q_8)
% in TeX

G:=Group("C12.88(C2xQ8)");
// GroupNames label

G:=SmallGroup(192,678);
// by ID

G=gap.SmallGroup(192,678);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,422,387,58,136,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^4=1,b^2=a^6,d^2=a^9*c^2,a*b=b*a,c*a*c^-1=a^5,a*d=d*a,c*b*c^-1=d*b*d^-1=a^6*b,d*c*d^-1=a^6*c^-1>;
// generators/relations

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