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G = C2×C232D6order 192 = 26·3

Direct product of C2 and C232D6

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

Aliases: C2×C232D6, C2410D6, C62C22≀C2, (C2×D4)⋊35D6, D612(C2×D4), (S3×C24)⋊3C2, (C2×C12)⋊9C23, (C22×D4)⋊8S3, (C22×C4)⋊29D6, (C22×C6)⋊11D4, D6⋊C471C22, (C6×D4)⋊55C22, (C22×S3)⋊15D4, C237(C3⋊D4), (C22×C6)⋊5C23, C234(C22×S3), (C2×C6).294C24, (C23×C6)⋊12C22, (C2×Dic3)⋊3C23, C22.146(S3×D4), C6.141(C22×D4), (S3×C23)⋊21C22, (C22×C12)⋊43C22, C6.D460C22, C22.307(S3×C23), (C22×S3).238C23, (C22×Dic3)⋊32C22, (D4×C2×C6)⋊15C2, (C2×C6)⋊7(C2×D4), C33(C2×C22≀C2), C2.101(C2×S3×D4), (C2×D6⋊C4)⋊41C2, (C2×C4)⋊4(C22×S3), C224(C2×C3⋊D4), (C22×C3⋊D4)⋊12C2, (C2×C3⋊D4)⋊43C22, C2.14(C22×C3⋊D4), (C2×C6.D4)⋊27C2, SmallGroup(192,1358)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C2×C232D6
C1C3C6C2×C6C22×S3S3×C23S3×C24 — C2×C232D6
C3C2×C6 — C2×C232D6
C1C23C22×D4

Generators and relations for C2×C232D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1960 in 662 conjugacy classes, 143 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×14], C3, C4 [×6], C22, C22 [×10], C22 [×86], S3 [×8], C6, C6 [×6], C6 [×6], C2×C4 [×2], C2×C4 [×10], D4 [×24], C23, C23 [×8], C23 [×86], Dic3 [×4], C12 [×2], D6 [×8], D6 [×56], C2×C6, C2×C6 [×10], C2×C6 [×22], C22⋊C4 [×12], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×20], C24 [×2], C24 [×18], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×16], C2×C12 [×2], C2×C12 [×2], C3×D4 [×8], C22×S3 [×12], C22×S3 [×64], C22×C6, C22×C6 [×8], C22×C6 [×10], C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4, C22×D4 [×2], C25, D6⋊C4 [×8], C6.D4 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×8], C2×C3⋊D4 [×8], C22×C12, C6×D4 [×4], C6×D4 [×4], S3×C23 [×6], S3×C23 [×12], C23×C6 [×2], C2×C22≀C2, C2×D6⋊C4 [×2], C232D6 [×8], C2×C6.D4, C22×C3⋊D4 [×2], D4×C2×C6, S3×C24, C2×C232D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×12], C23 [×15], D6 [×7], C2×D4 [×18], C24, C3⋊D4 [×4], C22×S3 [×7], C22≀C2 [×4], C22×D4 [×3], S3×D4 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C22≀C2, C232D6 [×4], C2×S3×D4 [×2], C22×C3⋊D4, C2×C232D6

Smallest permutation representation of C2×C232D6
On 48 points
Generators in S48
(1 32)(2 33)(3 34)(4 35)(5 36)(6 31)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)(13 41)(14 42)(15 37)(16 38)(17 39)(18 40)(25 44)(26 45)(27 46)(28 47)(29 48)(30 43)
(1 44)(2 12)(3 46)(4 8)(5 48)(6 10)(7 14)(9 16)(11 18)(13 45)(15 47)(17 43)(19 31)(20 40)(21 33)(22 42)(23 35)(24 38)(25 32)(26 41)(27 34)(28 37)(29 36)(30 39)
(1 4)(2 5)(3 6)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 16)(14 17)(15 18)(19 27)(20 28)(21 29)(22 30)(23 25)(24 26)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)
(1 18)(2 13)(3 14)(4 15)(5 16)(6 17)(7 46)(8 47)(9 48)(10 43)(11 44)(12 45)(19 30)(20 25)(21 26)(22 27)(23 28)(24 29)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)
(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 14)(2 13)(3 18)(4 17)(5 16)(6 15)(7 47)(8 46)(9 45)(10 44)(11 43)(12 48)(19 25)(20 30)(21 29)(22 28)(23 27)(24 26)(31 37)(32 42)(33 41)(34 40)(35 39)(36 38)

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

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

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

48 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N···2U 3 4A4B4C4D4E4F6A···6G6H···6O12A12B12C12D
order12···22222222···234444446···66···612121212
size11···12222446···6244121212122···24···44444

48 irreducible representations

dim111111122222224
type++++++++++++++
imageC1C2C2C2C2C2C2S3D4D4D6D6D6C3⋊D4S3×D4
kernelC2×C232D6C2×D6⋊C4C232D6C2×C6.D4C22×C3⋊D4D4×C2×C6S3×C24C22×D4C22×S3C22×C6C22×C4C2×D4C24C23C22
# reps128121118414284

Matrix representation of C2×C232D6 in GL5(𝔽13)

120000
01000
00100
00010
00001
,
120000
001200
012000
000119
00042
,
10000
012000
001200
000120
000012
,
10000
012000
001200
00010
00001
,
120000
012000
00100
000012
00011
,
10000
012000
001200
00001
00010

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

C2×C232D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes_2D_6
% in TeX

G:=Group("C2xC2^3:2D6");
// GroupNames label

G:=SmallGroup(192,1358);
// by ID

G=gap.SmallGroup(192,1358);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations

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