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G = C3×C22.33C24  order 192 = 26·3

Direct product of C3 and C22.33C24

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

Aliases: C3×C22.33C24, C6.1552+ (1+4), C6.1132- (1+4), (C4×D4)⋊11C6, C22⋊Q8⋊8C6, (D4×C12)⋊40C2, C4⋊D4.8C6, C42.C2⋊4C6, C42⋊2C2⋊4C6, C42.37(C2×C6), (C2×C6).359C24, C22.D4⋊5C6, (C2×C12).668C23, (C4×C12).278C22, (C6×D4).217C22, C23.42(C22×C6), (C22×C6).94C23, C22.33(C23×C6), (C6×Q8).181C22, C2.5(C3×2- (1+4)), C2.7(C3×2+ (1+4)), (C22×C12).38C22, (C2×C4⋊C4)⋊19C6, (C6×C4⋊C4)⋊46C2, C4⋊C4.68(C2×C6), C2.16(C6×C4○D4), (C2×D4).31(C2×C6), C6.235(C2×C4○D4), (C3×C22⋊Q8)⋊35C2, (C2×Q8).26(C2×C6), C22.5(C3×C4○D4), (C2×C6).53(C4○D4), (C3×C4⋊D4).18C2, C22⋊C4.15(C2×C6), (C22×C4).64(C2×C6), (C2×C4).26(C22×C6), (C3×C42.C2)⋊21C2, (C3×C42⋊2C2)⋊13C2, (C3×C4⋊C4).392C22, (C3×C22.D4)⋊24C2, (C3×C22⋊C4).85C22, SmallGroup(192,1428)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3×C22.33C24
Chief series
C1 — C2 — C22 — C2×C6 — C22×C6 — C3×C22⋊C4 — C3×C42⋊2C2 — C3×C22.33C24
Lower central
C1 — C22 — C3×C22.33C24
Upper central
C1 — C2×C6 — C3×C22.33C24

Subgroups: 322 in 218 conjugacy classes, 146 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×12], C22, C22 [×2], C22 [×8], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×10], C2×C4 [×6], D4 [×5], Q8, C23, C23 [×2], C12 [×12], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×12], C22×C4 [×3], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×C12 [×2], C2×C12 [×10], C2×C12 [×6], C3×D4 [×5], C3×Q8, C22×C6, C22×C6 [×2], C2×C4⋊C4, C4×D4 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×4], C42.C2 [×2], C42⋊2C2 [×2], C4×C12 [×2], C3×C22⋊C4 [×10], C3×C4⋊C4 [×2], C3×C4⋊C4 [×12], C22×C12 [×3], C22×C12 [×2], C6×D4, C6×D4 [×2], C6×Q8, C22.33C24, C6×C4⋊C4, D4×C12 [×2], C3×C4⋊D4, C3×C22⋊Q8, C3×C22⋊Q8 [×2], C3×C22.D4 [×4], C3×C42.C2 [×2], C3×C42⋊2C2 [×2], C3×C22.33C24

Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], C23 [×15], C2×C6 [×35], C4○D4 [×2], C24, C22×C6 [×15], C2×C4○D4, 2+ (1+4), 2- (1+4), C3×C4○D4 [×2], C23×C6, C22.33C24, C6×C4○D4, C3×2+ (1+4), C3×2- (1+4), C3×C22.33C24

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

Smallest permutation representation
►On 96 points
Generators in S96
(1 59 11)(2 60 12)(3 57 9)(4 58 10)(5 54 26)(6 55 27)(7 56 28)(8 53 25)(13 17 61)(14 18 62)(15 19 63)(16 20 64)(21 65 69)(22 66 70)(23 67 71)(24 68 72)(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 51 93)(46 52 94)(47 49 95)(48 50 96)

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

(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)

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

100000
010000
009000
000900
000090
000009
,
100000
010000
0012000
0001200
0000120
0000012
,
1200000
0120000
001000
000100
000010
000001
,
100000
0120000
001000
0001200
000010
0000012
,
500000
050000
0001106
0011060
000602
006020
,
010000
100000
000010
000001
0012000
0001200
,
100000
010000
000100
001000
000001
000010

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

66 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B4C4D4E···4N6A···6F6G6H6I6J6K6L6M6N12A···12H12I···12AB
order122222223344444···46···66666666612···1212···12
size111122441122224···41···1222244442···24···4

66 irreducible representations

dim1111111111111111224444
type+++++++++-
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6C4○D4C3×C4○D42+ (1+4)2- (1+4)C3×2+ (1+4)C3×2- (1+4)
kernelC3×C22.33C24C6×C4⋊C4D4×C12C3×C4⋊D4C3×C22⋊Q8C3×C22.D4C3×C42.C2C3×C42⋊2C2C22.33C24C2×C4⋊C4C4×D4C4⋊D4C22⋊Q8C22.D4C42.C2C42⋊2C2C2×C6C22C6C6C2C2
# reps1121342222426844481122

In GAP, Magma, Sage, TeX 

C_3\times C_2^2._{33}C_2^4
 
% in TeX
 
G:=Group("C3xC2^2.33C2^4");
 
// GroupNames label
 
G:=SmallGroup(192,1428);
 
// by ID
 
G=gap.SmallGroup(192,1428);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,680,2102,555,268,1571]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=g^2=1,e^2=c,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=g*d*g=b*d=d*b,f*e*f^-1=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,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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