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G = C2×C244S3order 192 = 26·3

Direct product of C2 and C244S3

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C244S3, C254S3, C2416D6, C63C22≀C2, (C24×C6)⋊3C2, (C22×C6)⋊18D4, C238(C3⋊D4), (C2×C6).322C24, (C22×S3)⋊3C23, (C23×C6)⋊18C22, (C2×Dic3)⋊4C23, C6.176(C22×D4), (S3×C23)⋊16C22, C6.D466C22, C22.330(S3×C23), C23.251(C22×S3), (C22×C6).429C23, (C22×Dic3)⋊37C22, C34(C2×C22≀C2), (C2×C6)⋊16(C2×D4), C225(C2×C3⋊D4), (C22×C3⋊D4)⋊18C2, (C2×C3⋊D4)⋊49C22, C2.48(C22×C3⋊D4), (C2×C6.D4)⋊32C2, SmallGroup(192,1399)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C2×C244S3
C1C3C6C2×C6C22×S3S3×C23C22×C3⋊D4 — C2×C244S3
C3C2×C6 — C2×C244S3
C1C23C25

Generators and relations for C2×C244S3
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, gbg=be=eb, bf=fb, gcg=cd=dc, ce=ec, cf=fc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg=f-1 >

Subgroups: 1528 in 662 conjugacy classes, 159 normal (10 characteristic)
C1, C2 [×7], C2 [×14], C3, C4 [×6], C22, C22 [×18], C22 [×78], S3 [×2], C6 [×7], C6 [×12], C2×C4 [×12], D4 [×24], C23, C23 [×18], C23 [×76], Dic3 [×6], D6 [×10], C2×C6, C2×C6 [×18], C2×C6 [×68], C22⋊C4 [×12], C22×C4 [×3], C2×D4 [×24], C24 [×7], C24 [×13], C2×Dic3 [×6], C2×Dic3 [×6], C3⋊D4 [×24], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6 [×18], C22×C6 [×68], C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4 [×3], C25, C6.D4 [×12], C22×Dic3 [×3], C2×C3⋊D4 [×12], C2×C3⋊D4 [×12], S3×C23, C23×C6 [×7], C23×C6 [×12], C2×C22≀C2, C2×C6.D4 [×3], C244S3 [×8], C22×C3⋊D4 [×3], C24×C6, C2×C244S3
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×12], C23 [×15], D6 [×7], C2×D4 [×18], C24, C3⋊D4 [×12], C22×S3 [×7], C22≀C2 [×4], C22×D4 [×3], C2×C3⋊D4 [×18], S3×C23, C2×C22≀C2, C244S3 [×4], C22×C3⋊D4 [×3], C2×C244S3

Smallest permutation representation of C2×C244S3
On 48 points
Generators in S48
(1 14)(2 15)(3 13)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(1 20)(2 21)(3 19)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 46)(26 47)(27 48)(28 43)(29 44)(30 45)(31 40)(32 41)(33 42)(34 37)(35 38)(36 39)
(1 14)(2 15)(3 13)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)
(1 8)(2 9)(3 7)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)
(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)
(1 40)(2 42)(3 41)(4 38)(5 37)(6 39)(7 47)(8 46)(9 48)(10 44)(11 43)(12 45)(13 29)(14 28)(15 30)(16 26)(17 25)(18 27)(19 35)(20 34)(21 36)(22 32)(23 31)(24 33)

G:=sub<Sym(48)| (1,14)(2,15)(3,13)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,20)(2,21)(3,19)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,46)(26,47)(27,48)(28,43)(29,44)(30,45)(31,40)(32,41)(33,42)(34,37)(35,38)(36,39), (1,14)(2,15)(3,13)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48), (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), (1,40)(2,42)(3,41)(4,38)(5,37)(6,39)(7,47)(8,46)(9,48)(10,44)(11,43)(12,45)(13,29)(14,28)(15,30)(16,26)(17,25)(18,27)(19,35)(20,34)(21,36)(22,32)(23,31)(24,33)>;

G:=Group( (1,14)(2,15)(3,13)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (1,20)(2,21)(3,19)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,46)(26,47)(27,48)(28,43)(29,44)(30,45)(31,40)(32,41)(33,42)(34,37)(35,38)(36,39), (1,14)(2,15)(3,13)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42), (1,8)(2,9)(3,7)(4,10)(5,11)(6,12)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48), (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), (1,40)(2,42)(3,41)(4,38)(5,37)(6,39)(7,47)(8,46)(9,48)(10,44)(11,43)(12,45)(13,29)(14,28)(15,30)(16,26)(17,25)(18,27)(19,35)(20,34)(21,36)(22,32)(23,31)(24,33) );

G=PermutationGroup([(1,14),(2,15),(3,13),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(1,20),(2,21),(3,19),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,46),(26,47),(27,48),(28,43),(29,44),(30,45),(31,40),(32,41),(33,42),(34,37),(35,38),(36,39)], [(1,14),(2,15),(3,13),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42)], [(1,8),(2,9),(3,7),(4,10),(5,11),(6,12),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48)], [(1,5),(2,6),(3,4),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48)], [(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)], [(1,40),(2,42),(3,41),(4,38),(5,37),(6,39),(7,47),(8,46),(9,48),(10,44),(11,43),(12,45),(13,29),(14,28),(15,30),(16,26),(17,25),(18,27),(19,35),(20,34),(21,36),(22,32),(23,31),(24,33)])

60 conjugacy classes

class 1 2A···2G2H···2S2T2U 3 4A···4F6A···6AE
order12···22···22234···46···6
size11···12···21212212···122···2

60 irreducible representations

dim111112222
type++++++++
imageC1C2C2C2C2S3D4D6C3⋊D4
kernelC2×C244S3C2×C6.D4C244S3C22×C3⋊D4C24×C6C25C22×C6C24C23
# reps13831112724

Matrix representation of C2×C244S3 in GL6(𝔽13)

100000
010000
0012000
0001200
0000120
0000012
,
100000
0120000
0012000
000100
000010
000001
,
100000
010000
0012000
0001200
0000120
000001
,
100000
010000
001000
000100
0000120
0000012
,
1200000
0120000
0012000
0001200
000010
000001
,
300000
090000
009000
000300
000010
000001
,
0120000
1200000
000100
001000
0000012
0000120

G:=sub<GL(6,GF(13))| [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],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

C2×C244S3 in GAP, Magma, Sage, TeX

C_2\times C_2^4\rtimes_4S_3
% in TeX

G:=Group("C2xC2^4:4S3");
// GroupNames label

G:=SmallGroup(192,1399);
// by ID

G=gap.SmallGroup(192,1399);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,675,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^3=g^2=1,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,b*d=d*b,g*b*g=b*e=e*b,b*f=f*b,g*c*g=c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,d*g=g*d,e*f=f*e,e*g=g*e,g*f*g=f^-1>;
// generators/relations

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