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G = C3×C22.M4(2)  order 192 = 26·3

Direct product of C3 and C22.M4(2)

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

Aliases: C3×C22.M4(2), (C2×C4)⋊C24, (C2×C12)⋊2C8, C22⋊C8.1C6, (C2×C12).442D4, C22.3(C2×C24), (C22×C12).5C4, (C22×C4).3C12, C6.21(C22⋊C8), C6.28(C23⋊C4), C23.27(C2×C12), (C2×C6).15M4(2), (C22×C12).2C22, C22.3(C3×M4(2)), C6.10(C4.10D4), (C2×C4⋊C4).1C6, (C6×C4⋊C4).28C2, (C2×C6).21(C2×C8), (C2×C4).92(C3×D4), C2.4(C3×C22⋊C8), C2.2(C3×C23⋊C4), (C3×C22⋊C8).3C2, (C22×C4).7(C2×C6), C2.1(C3×C4.10D4), (C22×C6).107(C2×C4), C22.24(C3×C22⋊C4), (C2×C6).119(C22⋊C4), SmallGroup(192,130)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3×C22.M4(2)
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C12 — C3×C22⋊C8 — C3×C22.M4(2)
Lower central
C1 — C2 — C22 — C3×C22.M4(2)
Upper central
C1 — C2×C6 — C22×C12 — C3×C22.M4(2)

Generators and relations for C3×C22.M4(2)
 G = < a,b,c,d,e | a3=b2=c2=d8=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd5 >

Subgroups: 138 in 78 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22⋊C8, C2×C4⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C22.M4(2), C3×C22⋊C8, C6×C4⋊C4, C3×C22.M4(2)
Quotients: C1, C2, C3, C4, C22, C6, C8, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C8, M4(2), C24, C2×C12, C3×D4, C22⋊C8, C23⋊C4, C4.10D4, C3×C22⋊C4, C2×C24, C3×M4(2), C22.M4(2), C3×C22⋊C8, C3×C23⋊C4, C3×C4.10D4, C3×C22.M4(2)

Smallest permutation representation of C3×C22.M4(2)
►On 96 points
Generators in S96
(1 74 91)(2 75 92)(3 76 93)(4 77 94)(5 78 95)(6 79 96)(7 80 89)(8 73 90)(9 65 81)(10 66 82)(11 67 83)(12 68 84)(13 69 85)(14 70 86)(15 71 87)(16 72 88)(17 51 27)(18 52 28)(19 53 29)(20 54 30)(21 55 31)(22 56 32)(23 49 25)(24 50 26)(33 41 59)(34 42 60)(35 43 61)(36 44 62)(37 45 63)(38 46 64)(39 47 57)(40 48 58)

(2 30)(4 32)(6 26)(8 28)(10 57)(12 59)(14 61)(16 63)(18 73)(20 75)(22 77)(24 79)(33 68)(35 70)(37 72)(39 66)(41 84)(43 86)(45 88)(47 82)(50 96)(52 90)(54 92)(56 94)

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

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

66 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F4G4H6A···6F6G6H6I6J8A···8H12A···12H12I···12P24A···24P
order12222233444444446···666668···812···1212···1224···24
size11112211222244441···122224···42···24···44···4

66 irreducible representations

dim111111111122224444
type+++++-
imageC1C2C2C3C4C6C6C8C12C24D4M4(2)C3×D4C3×M4(2)C23⋊C4C4.10D4C3×C23⋊C4C3×C4.10D4
kernelC3×C22.M4(2)C3×C22⋊C8C6×C4⋊C4C22.M4(2)C22×C12C22⋊C8C2×C4⋊C4C2×C12C22×C4C2×C4C2×C12C2×C6C2×C4C22C6C6C2C2
# reps1212442881622441122

Matrix representation of C3×C22.M4(2) ►in GL8(𝔽73)

10000000
01000000
006400000
000640000
00001000
00000100
00000010
00000001
,
10000000
01000000
007200000
000720000
00001000
00000100
000000720
00004627072
,
10000000
01000000
00100000
00010000
000072000
000007200
000000720
000000072
,
5971000000
214000000
000270000
002700000
00001261319
000046274271
0000327200
00000313934
,
01000000
10000000
007200000
00010000
0000723200
000041100
00001261319
00003903142

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,64,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,46,0,0,0,0,0,1,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72],[59,2,0,0,0,0,0,0,71,14,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,12,46,32,0,0,0,0,0,61,27,72,31,0,0,0,0,31,42,0,39,0,0,0,0,9,71,0,34],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,41,12,39,0,0,0,0,32,1,61,0,0,0,0,0,0,0,31,31,0,0,0,0,0,0,9,42] >;
 

C3×C22.M4(2) in GAP, Magma, Sage, TeX 

C_3\times C_2^2.M_4(2)
 
% in TeX
 
G:=Group("C3xC2^2.M4(2)");
 
// GroupNames label
 
G:=SmallGroup(192,130);
 
// by ID
 
G=gap.SmallGroup(192,130);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,1683,1271,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^8=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^5>;
 
// generators/relations
 

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