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G = C3×C22.57C24  order 192 = 26·3

Direct product of C3 and C22.57C24

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

Aliases: C3×C22.57C24, C6.1722+ 1+4, C6.1242- 1+4, C4⋊Q8⋊18C6, C22⋊Q8⋊20C6, C42.55(C2×C6), C42.C2⋊12C6, C4.4D4.9C6, C42⋊2C2⋊11C6, (C2×C6).383C24, (C4×C12).296C22, (C2×C12).684C23, (C6×D4).225C22, C23.26(C22×C6), C22.57(C23×C6), (C6×Q8).188C22, C22.D4.3C6, (C22×C6).109C23, C2.16(C3×2- 1+4), C2.24(C3×2+ 1+4), (C22×C12).463C22, (C3×C4⋊Q8)⋊39C2, C4⋊C4.35(C2×C6), (C2×D4).38(C2×C6), C22⋊C4.8(C2×C6), (C3×C22⋊Q8)⋊47C2, (C2×Q8).32(C2×C6), (C2×C4).43(C22×C6), (C22×C4).79(C2×C6), (C3×C42.C2)⋊29C2, (C3×C42⋊2C2)⋊20C2, (C3×C4⋊C4).252C22, (C3×C4.4D4).18C2, (C3×C22⋊C4).93C22, (C3×C22.D4).6C2, SmallGroup(192,1452)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3×C22.57C24
Chief series
C1 — C2 — C22 — C2×C6 — C2×C12 — C6×Q8 — C3×C4⋊Q8 — C3×C22.57C24
Lower central
C1 — C22 — C3×C22.57C24
Upper central
C1 — C2×C6 — C3×C22.57C24

Generators and relations for C3×C22.57C24
 G = < a,b,c,d,e,f,g | a3=b2=c2=g2=1, d2=e2=f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef-1=bce, fg=gf >

Subgroups: 282 in 196 conjugacy classes, 142 normal (18 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C4×C12, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, C6×Q8, C22.57C24, C3×C22⋊Q8, C3×C22.D4, C3×C4.4D4, C3×C42.C2, C3×C42⋊2C2, C3×C4⋊Q8, C3×C22.57C24
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C24, C22×C6, 2+ 1+4, 2- 1+4, C23×C6, C22.57C24, C3×2+ 1+4, C3×2- 1+4, C3×C22.57C24

Smallest permutation representation of C3×C22.57C24
►On 96 points
Generators in S96
(1 11 55)(2 12 56)(3 9 53)(4 10 54)(5 25 69)(6 26 70)(7 27 71)(8 28 72)(13 57 61)(14 58 62)(15 59 63)(16 60 64)(17 21 65)(18 22 66)(19 23 67)(20 24 68)(29 73 77)(30 74 78)(31 75 79)(32 76 80)(33 37 81)(34 38 82)(35 39 83)(36 40 84)(41 85 89)(42 86 90)(43 87 91)(44 88 92)(45 52 93)(46 49 94)(47 50 95)(48 51 96)

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

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

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

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

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

57 conjugacy classes

class 1 2A2B2C2D2E3A3B4A···4M6A···6F6G6H6I6J12A···12Z
order122222334···46···6666612···12
size111144114···41···144444···4

57 irreducible representations

dim111111111111114444
type++++++++-
imageC1C2C2C2C2C2C2C3C6C6C6C6C6C62+ 1+42- 1+4C3×2+ 1+4C3×2- 1+4
kernelC3×C22.57C24C3×C22⋊Q8C3×C22.D4C3×C4.4D4C3×C42.C2C3×C42⋊2C2C3×C4⋊Q8C22.57C24C22⋊Q8C22.D4C4.4D4C42.C2C42⋊2C2C4⋊Q8C6C6C2C2
# reps142124228424841224

Matrix representation of C3×C22.57C24 ►in GL9(𝔽13)

900000000
010000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
100000000
0120000000
0012000000
0001200000
0000120000
000001000
000000100
000000010
000000001
,
100000000
0120000000
0012000000
0001200000
0000120000
0000012000
0000001200
0000000120
0000000012
,
100000000
0123000000
081000000
0361100000
0375120000
00000105110
0000078011
000002638
00000111065
,
100000000
0871100000
0690110000
081560000
01210740000
000000500
000008000
000009505
000005480
,
100000000
0511000000
008000000
0655110000
047080000
000000100
000001000
0000069012
0000096120
,
1200000000
010000000
001000000
0871200000
0690120000
000001000
000000100
00000105120
0000078012

G:=sub<GL(9,GF(13))| [9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,0,12,8,3,3,0,0,0,0,0,3,1,6,7,0,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,10,12,0,0,0,0,0,0,0,0,0,10,7,2,11,0,0,0,0,0,5,8,6,10,0,0,0,0,0,11,0,3,6,0,0,0,0,0,0,11,8,5],[1,0,0,0,0,0,0,0,0,0,8,6,8,12,0,0,0,0,0,7,9,1,10,0,0,0,0,0,11,0,5,7,0,0,0,0,0,0,11,6,4,0,0,0,0,0,0,0,0,0,0,8,9,5,0,0,0,0,0,5,0,5,4,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,5,0],[1,0,0,0,0,0,0,0,0,0,5,0,6,4,0,0,0,0,0,11,8,5,7,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,11,8,0,0,0,0,0,0,0,0,0,0,1,6,9,0,0,0,0,0,1,0,9,6,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0],[12,0,0,0,0,0,0,0,0,0,1,0,8,6,0,0,0,0,0,0,1,7,9,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,10,7,0,0,0,0,0,0,1,5,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12] >;
 

C3×C22.57C24 in GAP, Magma, Sage, TeX 

C_3\times C_2^2._{57}C_2^4
 
% in TeX
 
G:=Group("C3xC2^2.57C2^4");
 
// GroupNames label
 
G:=SmallGroup(192,1452);
 
// by ID
 
G=gap.SmallGroup(192,1452);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,336,701,344,2102,1563,268,4259,794]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=g^2=1,d^2=e^2=f^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,g*d*g=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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