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G = C3×C23.32C23  order 192 = 26·3

Direct product of C3 and C23.32C23

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

Aliases: C3×C23.32C23, C6.1092- (1+4), (C4×Q8)⋊9C6, (C6×Q8)⋊19C4, (C2×Q8)⋊11C12, (Q8×C12)⋊25C2, C2.8(C23×C12), C6.60(C23×C4), C42.32(C2×C6), Q8.13(C2×C12), C4.20(C22×C12), (C2×C6).339C24, (C22×Q8).14C6, (C2×C12).710C23, C12.165(C22×C4), C42⋊C2.10C6, (C4×C12).275C22, C22.12(C23×C6), C23.35(C22×C6), (C6×Q8).284C22, C2.1(C3×2- (1+4)), (C22×C6).255C23, C22.11(C22×C12), (C22×C12).442C22, (Q8×C2×C6).17C2, C4⋊C4.82(C2×C6), (C2×C4).31(C2×C12), (C3×Q8).32(C2×C4), (C2×Q8).84(C2×C6), (C2×C12).203(C2×C4), C22⋊C4.29(C2×C6), (C22×C4).57(C2×C6), (C3×C4⋊C4).407C22, (C2×C6).165(C22×C4), (C2×C4).135(C22×C6), (C3×C42⋊C2).24C2, (C3×C22⋊C4).160C22, SmallGroup(192,1408)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — C3×C23.32C23
Chief series
C1 — C2 — C22 — C2×C6 — C2×C12 — C3×C4⋊C4 — Q8×C12 — C3×C23.32C23
Lower central
C1 — C2 — C3×C23.32C23
Upper central
C1 — C2×C6 — C3×C23.32C23

Subgroups: 290 in 266 conjugacy classes, 242 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×12], C4 [×8], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×26], Q8 [×16], C23, C12 [×12], C12 [×8], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×12], C22⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×3], C2×Q8 [×12], C2×C12 [×26], C3×Q8 [×16], C22×C6, C42⋊C2 [×6], C4×Q8 [×8], C22×Q8, C4×C12 [×12], C3×C22⋊C4 [×4], C3×C4⋊C4 [×12], C22×C12 [×3], C6×Q8 [×12], C23.32C23, C3×C42⋊C2 [×6], Q8×C12 [×8], Q8×C2×C6, C3×C23.32C23

Quotients:
C1, C2 [×15], C3, C4 [×8], C22 [×35], C6 [×15], C2×C4 [×28], C23 [×15], C12 [×8], C2×C6 [×35], C22×C4 [×14], C24, C2×C12 [×28], C22×C6 [×15], C23×C4, 2- (1+4) [×2], C22×C12 [×14], C23×C6, C23.32C23, C23×C12, C3×2- (1+4) [×2], C3×C23.32C23

Generators and relations
 G = < a,b,c,d,e,f,g | a3=b2=c2=d2=1, e2=d, f2=g2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ebe-1=bc=cb, bd=db, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, gfg-1=cf=fc, cg=gc, de=ed, df=fd, dg=gd, eg=ge >

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

(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)

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

Matrix representation ►G ⊆ GL6(𝔽13)

100000
090000
001000
000100
000010
000001
,
100000
0120000
001000
000100
0000120
0000012
,
100000
010000
0012000
0001200
0000120
0000012
,
1200000
010000
001000
000100
000010
000001
,
800000
010000
000010
000001
001000
000100
,
1200000
010000
000800
008000
000005
000050
,
100000
010000
000100
0012000
000001
0000120

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;
 

102 conjugacy classes

class 1 2A2B2C2D2E3A3B4A···4AB6A···6F6G6H6I6J12A···12BD
order122222334···46···6666612···12
size111122112···21···122222···2

102 irreducible representations

dim111111111144
type++++-
imageC1C2C2C2C3C4C6C6C6C122- (1+4)C3×2- (1+4)
kernelC3×C23.32C23C3×C42⋊C2Q8×C12Q8×C2×C6C23.32C23C6×Q8C42⋊C2C4×Q8C22×Q8C2×Q8C6C2
# reps1681216121623224

In GAP, Magma, Sage, TeX 

C_3\times C_2^3._{32}C_2^3
 
% in TeX
 
G:=Group("C3xC2^3.32C2^3");
 
// GroupNames label
 
G:=SmallGroup(192,1408);
 
// by ID
 
G=gap.SmallGroup(192,1408);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,344,555,268,1571]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=1,e^2=d,f^2=g^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*b*e^-1=b*c=c*b,b*d=d*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,g*f*g^-1=c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
 
// generators/relations
 

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