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G = C6×C22⋊C4order 96 = 25·3

Direct product of C6 and C22⋊C4

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

Aliases: C6×C22⋊C4, C233C12, C24.2C6, C2.1(C6×D4), (C22×C4)⋊3C6, (C22×C6)⋊3C4, (C2×C6).50D4, C6.64(C2×D4), (C22×C12)⋊3C2, C223(C2×C12), C23.8(C2×C6), (C23×C6).1C2, (C2×C12)⋊11C22, C2.1(C22×C12), C6.29(C22×C4), (C2×C6).70C23, C22.12(C3×D4), C22.4(C22×C6), (C22×C6).24C22, (C2×C4)⋊3(C2×C6), (C2×C6)⋊7(C2×C4), SmallGroup(96,162)

Series: Derived Chief Lower central Upper central

C1C2 — C6×C22⋊C4
C1C2C22C2×C6C2×C12C3×C22⋊C4 — C6×C22⋊C4
C1C2 — C6×C22⋊C4
C1C22×C6 — C6×C22⋊C4

Generators and relations for C6×C22⋊C4
 G = < a,b,c,d | a6=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 188 in 132 conjugacy classes, 76 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×4], C22, C22 [×10], C22 [×12], C6, C6 [×6], C6 [×4], C2×C4 [×4], C2×C4 [×4], C23, C23 [×6], C23 [×4], C12 [×4], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×4], C22×C4 [×2], C24, C2×C12 [×4], C2×C12 [×4], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4, C3×C22⋊C4 [×4], C22×C12 [×2], C23×C6, C6×C22⋊C4
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C2×C4 [×6], D4 [×4], C23, C12 [×4], C2×C6 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C12 [×6], C3×D4 [×4], C22×C6, C2×C22⋊C4, C3×C22⋊C4 [×4], C22×C12, C6×D4 [×2], C6×C22⋊C4

Smallest permutation representation of C6×C22⋊C4
On 48 points
Generators in S48
(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 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 20)(14 21)(15 22)(16 23)(17 24)(18 19)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)
(1 39)(2 40)(3 41)(4 42)(5 37)(6 38)(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 11 26 17)(2 12 27 18)(3 7 28 13)(4 8 29 14)(5 9 30 15)(6 10 25 16)(19 32 48 40)(20 33 43 41)(21 34 44 42)(22 35 45 37)(23 36 46 38)(24 31 47 39)

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

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

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

C6×C22⋊C4 is a maximal subgroup of
C24.3Dic3  C24.12D6  C24.13D6  C24.55D6  C24.56D6  C24.14D6  C24.15D6  C24.57D6  C232Dic6  C24.17D6  C24.18D6  C24.58D6  C24.19D6  C24.20D6  C24.21D6  C24.59D6  C24.23D6  C24.24D6  C24.60D6  C24.25D6  C233D12  C24.27D6  C233Dic6  C24.35D6  C24.38D6  C234D12  C24.41D6  C24.42D6  D4×C2×C12

60 conjugacy classes

class 1 2A···2G2H2I2J2K3A3B4A···4H6A···6N6O···6V12A···12P
order12···22222334···46···66···612···12
size11···12222112···21···12···22···2

60 irreducible representations

dim111111111122
type+++++
imageC1C2C2C2C3C4C6C6C6C12D4C3×D4
kernelC6×C22⋊C4C3×C22⋊C4C22×C12C23×C6C2×C22⋊C4C22×C6C22⋊C4C22×C4C24C23C2×C6C22
# reps1421288421648

Matrix representation of C6×C22⋊C4 in GL4(𝔽13) generated by

12000
01000
0010
0001
,
1000
01200
0010
00012
,
1000
0100
00120
00012
,
5000
0100
00012
00120
G:=sub<GL(4,GF(13))| [12,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,12,0,0,0,0,1,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[5,0,0,0,0,1,0,0,0,0,0,12,0,0,12,0] >;

C6×C22⋊C4 in GAP, Magma, Sage, TeX

C_6\times C_2^2\rtimes C_4
% in TeX

G:=Group("C6xC2^2:C4");
// GroupNames label

G:=SmallGroup(96,162);
// by ID

G=gap.SmallGroup(96,162);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

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