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## G = C4.(C2×D12)  order 192 = 26·3

### 14th non-split extension by C4 of C2×D12 acting via C2×D12/C22×S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4.(C2×D12)
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C2×C4○D12 — C4.(C2×D12)
 Lower central C3 — C6 — C12 — C4.(C2×D12)
 Upper central C1 — C2×C4 — C22×C4 — C42⋊C2

Generators and relations for C4.(C2×D12)
G = < a,b,c,d | a4=b2=c12=1, d2=a, ab=ba, cac-1=a-1, ad=da, cbc-1=a2b, bd=db, dcd-1=ac-1 >

Subgroups: 424 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, D4⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, C2×C3⋊C8, C2×C3⋊C8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C23.24D4, C6.D8, C6.SD16, C22×C3⋊C8, C3×C42⋊C2, C2×C4○D12, C4.(C2×D12)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C4○D8, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C23.24D4, C2×D6⋊C4, Q8.13D6, C4.(C2×D12)

Smallest permutation representation of C4.(C2×D12)
On 96 points
Generators in S96
(1 82 54 69)(2 70 55 83)(3 84 56 71)(4 72 57 73)(5 74 58 61)(6 62 59 75)(7 76 60 63)(8 64 49 77)(9 78 50 65)(10 66 51 79)(11 80 52 67)(12 68 53 81)(13 92 39 31)(14 32 40 93)(15 94 41 33)(16 34 42 95)(17 96 43 35)(18 36 44 85)(19 86 45 25)(20 26 46 87)(21 88 47 27)(22 28 48 89)(23 90 37 29)(24 30 38 91)
(1 91)(2 31)(3 93)(4 33)(5 95)(6 35)(7 85)(8 25)(9 87)(10 27)(11 89)(12 29)(13 70)(14 84)(15 72)(16 74)(17 62)(18 76)(19 64)(20 78)(21 66)(22 80)(23 68)(24 82)(26 50)(28 52)(30 54)(32 56)(34 58)(36 60)(37 81)(38 69)(39 83)(40 71)(41 73)(42 61)(43 75)(44 63)(45 77)(46 65)(47 79)(48 67)(49 86)(51 88)(53 90)(55 92)(57 94)(59 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 68 82 53 54 81 69 12)(2 11 70 80 55 52 83 67)(3 66 84 51 56 79 71 10)(4 9 72 78 57 50 73 65)(5 64 74 49 58 77 61 8)(6 7 62 76 59 60 75 63)(13 22 92 28 39 48 31 89)(14 88 32 47 40 27 93 21)(15 20 94 26 41 46 33 87)(16 86 34 45 42 25 95 19)(17 18 96 36 43 44 35 85)(23 24 90 30 37 38 29 91)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 6A 6B 6C 6D 6E 8A ··· 8H 12A 12B 12C 12D 12E ··· 12N order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 ··· 8 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 12 12 2 1 1 1 1 2 2 4 4 4 4 12 12 2 2 2 4 4 6 ··· 6 2 2 2 2 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 C2 C4 S3 D4 D4 D6 D6 C4×S3 D12 C3⋊D4 C3⋊D4 C4○D8 Q8.13D6 kernel C4.(C2×D12) C6.D8 C6.SD16 C22×C3⋊C8 C3×C42⋊C2 C2×C4○D12 C4○D12 C42⋊C2 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 2 2 1 1 1 8 1 3 1 2 1 4 4 2 2 8 4

Matrix representation of C4.(C2×D12) in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 1 25 0 0 70 72
,
 1 0 0 0 0 1 0 0 0 0 27 18 0 0 65 46
,
 66 66 0 0 7 59 0 0 0 0 0 35 0 0 48 0
,
 7 59 0 0 66 66 0 0 0 0 41 38 0 0 48 0
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,1,70,0,0,25,72],[1,0,0,0,0,1,0,0,0,0,27,65,0,0,18,46],[66,7,0,0,66,59,0,0,0,0,0,48,0,0,35,0],[7,66,0,0,59,66,0,0,0,0,41,48,0,0,38,0] >;

C4.(C2×D12) in GAP, Magma, Sage, TeX

C_4.(C_2\times D_{12})
% in TeX

G:=Group("C4.(C2xD12)");
// GroupNames label

G:=SmallGroup(192,561);
// by ID

G=gap.SmallGroup(192,561);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,422,58,1684,438,102,6278]);
// Polycyclic

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

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