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## G = C2×C24.C4order 192 = 26·3

### Direct product of C2 and C24.C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×C24.C4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4.Dic3 — C2×C4.Dic3 — C2×C24.C4
 Lower central C3 — C6 — C12 — C2×C24.C4
 Upper central C1 — C2×C4 — C22×C4 — C22×C8

Generators and relations for C2×C24.C4
G = < a,b,c,d | a2=b8=1, c6=b4, d2=b4c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 184 in 106 conjugacy classes, 71 normal (41 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C8, C8, C2×C4, C23, C12, C2×C6, C2×C6, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C2×C12, C22×C6, C8.C4, C22×C8, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4.Dic3, C2×C24, C2×C24, C22×C12, C2×C8.C4, C24.C4, C2×C4.Dic3, C22×C24, C2×C24.C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, Dic3, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, Dic6, D12, C2×Dic3, C22×S3, C8.C4, C2×C4⋊C4, C4⋊Dic3, C2×Dic6, C2×D12, C22×Dic3, C2×C8.C4, C24.C4, C2×C4⋊Dic3, C2×C24.C4

Smallest permutation representation of C2×C24.C4
On 96 points
Generators in S96
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 85)(44 86)(45 87)(46 88)(47 89)(48 90)(49 76)(50 77)(51 78)(52 79)(53 80)(54 81)(55 82)(56 83)(57 84)(58 73)(59 74)(60 75)
(1 74 34 50 7 80 28 56)(2 75 35 51 8 81 29 57)(3 76 36 52 9 82 30 58)(4 77 25 53 10 83 31 59)(5 78 26 54 11 84 32 60)(6 79 27 55 12 73 33 49)(13 48 70 87 19 42 64 93)(14 37 71 88 20 43 65 94)(15 38 72 89 21 44 66 95)(16 39 61 90 22 45 67 96)(17 40 62 91 23 46 68 85)(18 41 63 92 24 47 69 86)
(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 44 10 41 7 38 4 47)(2 37 11 46 8 43 5 40)(3 42 12 39 9 48 6 45)(13 79 22 76 19 73 16 82)(14 84 23 81 20 78 17 75)(15 77 24 74 21 83 18 80)(25 92 34 89 31 86 28 95)(26 85 35 94 32 91 29 88)(27 90 36 87 33 96 30 93)(49 67 58 64 55 61 52 70)(50 72 59 69 56 66 53 63)(51 65 60 62 57 71 54 68)

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + - - - + + - + - image C1 C2 C2 C2 C4 S3 D4 Q8 Q8 Dic3 D6 D6 Dic6 D12 Dic6 C8.C4 C24.C4 kernel C2×C24.C4 C24.C4 C2×C4.Dic3 C22×C24 C2×C24 C22×C8 C2×C12 C2×C12 C22×C6 C2×C8 C2×C8 C22×C4 C2×C4 C2×C4 C23 C6 C2 # reps 1 4 2 1 8 1 2 1 1 4 2 1 2 4 2 8 16

Matrix representation of C2×C24.C4 in GL4(𝔽73) generated by

 1 0 0 0 0 1 0 0 0 0 72 0 0 0 0 72
,
 63 0 0 0 0 51 0 0 0 0 66 14 0 0 59 7
,
 46 0 0 0 0 46 0 0 0 0 0 72 0 0 1 72
,
 0 22 0 0 22 0 0 0 0 0 10 41 0 0 51 63
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[63,0,0,0,0,51,0,0,0,0,66,59,0,0,14,7],[46,0,0,0,0,46,0,0,0,0,0,1,0,0,72,72],[0,22,0,0,22,0,0,0,0,0,10,51,0,0,41,63] >;

C2×C24.C4 in GAP, Magma, Sage, TeX

C_2\times C_{24}.C_4
% in TeX

G:=Group("C2xC24.C4");
// GroupNames label

G:=SmallGroup(192,666);
// by ID

G=gap.SmallGroup(192,666);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,100,136,1684,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^6=b^4,d^2=b^4*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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