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G = C3×M4(2)⋊C4order 192 = 26·3

Direct product of C3 and M4(2)⋊C4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C3×M4(2)⋊C4
 Chief series C1 — C2 — C22 — C2×C4 — C2×C12 — C3×C4⋊C4 — C3×C4.Q8 — C3×M4(2)⋊C4
 Lower central C1 — C2 — C4 — C3×M4(2)⋊C4
 Upper central C1 — C2×C6 — C22×C12 — C3×M4(2)⋊C4

Generators and relations for C3×M4(2)⋊C4
G = < a,b,c,d | a3=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, cd=dc >

Subgroups: 178 in 118 conjugacy classes, 82 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C22×C12, C22×C12, M4(2)⋊C4, C3×C4.Q8, C3×C2.D8, C6×C4⋊C4, C3×C42⋊C2, C6×M4(2), C3×M4(2)⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C8⋊C22, C8.C22, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, M4(2)⋊C4, C6×C4⋊C4, C3×C8⋊C22, C3×C8.C22, C3×M4(2)⋊C4

Smallest permutation representation of C3×M4(2)⋊C4
On 96 points
Generators in S96
(1 53 43)(2 54 44)(3 55 45)(4 56 46)(5 49 47)(6 50 48)(7 51 41)(8 52 42)(9 40 22)(10 33 23)(11 34 24)(12 35 17)(13 36 18)(14 37 19)(15 38 20)(16 39 21)(25 90 68)(26 91 69)(27 92 70)(28 93 71)(29 94 72)(30 95 65)(31 96 66)(32 89 67)(57 79 87)(58 80 88)(59 73 81)(60 74 82)(61 75 83)(62 76 84)(63 77 85)(64 78 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)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(66 70)(68 72)(73 77)(75 79)(81 85)(83 87)(90 94)(92 96)
(1 78 39 71)(2 77 40 70)(3 76 33 69)(4 75 34 68)(5 74 35 67)(6 73 36 66)(7 80 37 65)(8 79 38 72)(9 92 44 63)(10 91 45 62)(11 90 46 61)(12 89 47 60)(13 96 48 59)(14 95 41 58)(15 94 42 57)(16 93 43 64)(17 32 49 82)(18 31 50 81)(19 30 51 88)(20 29 52 87)(21 28 53 86)(22 27 54 85)(23 26 55 84)(24 25 56 83)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C3 C4 C6 C6 C6 C6 C6 C12 D4 Q8 D4 C3×D4 C3×Q8 C3×D4 C8⋊C22 C8.C22 C3×C8⋊C22 C3×C8.C22 kernel C3×M4(2)⋊C4 C3×C4.Q8 C3×C2.D8 C6×C4⋊C4 C3×C42⋊C2 C6×M4(2) M4(2)⋊C4 C3×M4(2) C4.Q8 C2.D8 C2×C4⋊C4 C42⋊C2 C2×M4(2) M4(2) C2×C12 C2×C12 C22×C6 C2×C4 C2×C4 C23 C6 C6 C2 C2 # reps 1 2 2 1 1 1 2 8 4 4 2 2 2 16 1 2 1 2 4 2 1 1 2 2

Matrix representation of C3×M4(2)⋊C4 in GL6(𝔽73)

 64 0 0 0 0 0 0 64 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 0 72 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 72 0 0 0 0 1 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 54 21 0 0 0 0 21 19 0 0 0 0 0 0 5 7 0 0 0 0 7 68 0 0 0 0 0 0 66 5 0 0 0 0 5 7

G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[54,21,0,0,0,0,21,19,0,0,0,0,0,0,5,7,0,0,0,0,7,68,0,0,0,0,0,0,66,5,0,0,0,0,5,7] >;

C3×M4(2)⋊C4 in GAP, Magma, Sage, TeX

C_3\times M_4(2)\rtimes C_4
% in TeX

G:=Group("C3xM4(2):C4");
// GroupNames label

G:=SmallGroup(192,861);
// by ID

G=gap.SmallGroup(192,861);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,1059,4204,172]);
// Polycyclic

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

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