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G = C2×C32⋊M4(2)  order 288 = 25·32

Direct product of C2 and C32⋊M4(2)

direct product, metabelian, soluble, monomial

Aliases: C2×C32⋊M4(2), (C6×C12).10C4, (C3×C6)⋊1M4(2), C62.14(C2×C4), C323(C2×M4(2)), C322C87C22, C3⋊Dic3.30C23, (C4×C3⋊S3).16C4, C4.12(C2×C32⋊C4), (C3×C12).19(C2×C4), (C2×C322C8)⋊7C2, (C2×C4).7(C32⋊C4), C2.4(C22×C32⋊C4), (C22×C3⋊S3).15C4, (C4×C3⋊S3).96C22, C3⋊Dic3.51(C2×C4), (C3×C6).25(C22×C4), C22.16(C2×C32⋊C4), (C2×C3⋊Dic3).173C22, (C2×C4×C3⋊S3).26C2, (C2×C3⋊S3).45(C2×C4), SmallGroup(288,930)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×C32⋊M4(2)
C1C32C3×C6C3⋊Dic3C322C8C2×C322C8 — C2×C32⋊M4(2)
C32C3×C6 — C2×C32⋊M4(2)
C1C22C2×C4

Generators and relations for C2×C32⋊M4(2)
 G = < a,b,c,d,e | a2=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=bc-1, ebe=b-1, dcd-1=b-1c-1, ece=c-1, ede=d5 >

Subgroups: 576 in 122 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C4 [×2], C4 [×2], C22, C22 [×4], S3 [×8], C6 [×6], C8 [×4], C2×C4, C2×C4 [×5], C23, C32, Dic3 [×4], C12 [×4], D6 [×12], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C3⋊S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×8], C2×Dic3 [×2], C2×C12 [×2], C22×S3 [×2], C2×M4(2), C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, S3×C2×C4 [×2], C322C8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C32⋊M4(2) [×4], C2×C322C8 [×2], C2×C4×C3⋊S3, C2×C32⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×2], C22×C4, C2×M4(2), C32⋊C4, C2×C32⋊C4 [×3], C32⋊M4(2) [×2], C22×C32⋊C4, C2×C32⋊M4(2)

Smallest permutation representation of C2×C32⋊M4(2)
On 48 points
Generators in S48
(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 46)(34 47)(35 48)(36 41)(37 42)(38 43)(39 44)(40 45)
(1 46 28)(3 30 48)(5 42 32)(7 26 44)(9 37 24)(11 18 39)(13 33 20)(15 22 35)
(1 46 28)(2 47 29)(3 30 48)(4 31 41)(5 42 32)(6 43 25)(7 26 44)(8 27 45)(9 37 24)(10 38 17)(11 18 39)(12 19 40)(13 33 20)(14 34 21)(15 22 35)(16 23 36)
(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)
(2 6)(4 8)(10 14)(12 16)(17 34)(18 39)(19 36)(20 33)(21 38)(22 35)(23 40)(24 37)(25 47)(26 44)(27 41)(28 46)(29 43)(30 48)(31 45)(32 42)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F6A···6F8A···8H12A···12H
order122222334444446···68···812···12
size11111818442299994···418···184···4

36 irreducible representations

dim111111124444
type+++++++
imageC1C2C2C2C4C4C4M4(2)C32⋊C4C2×C32⋊C4C2×C32⋊C4C32⋊M4(2)
kernelC2×C32⋊M4(2)C32⋊M4(2)C2×C322C8C2×C4×C3⋊S3C4×C3⋊S3C6×C12C22×C3⋊S3C3×C6C2×C4C4C22C2
# reps142142242428

Matrix representation of C2×C32⋊M4(2) in GL6(𝔽73)

100000
010000
0072000
0007200
0000720
0000072
,
100000
010000
00727200
001000
000010
000001
,
100000
010000
00727200
001000
00007272
000010
,
26460000
72470000
0000720
000011
0072000
0007200
,
100000
56720000
001000
00727200
000010
00007272

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[26,72,0,0,0,0,46,47,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,72,1,0,0,0,0,0,1,0,0],[1,56,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;

C2×C32⋊M4(2) in GAP, Magma, Sage, TeX

C_2\times C_3^2\rtimes M_4(2)
% in TeX

G:=Group("C2xC3^2:M4(2)");
// GroupNames label

G:=SmallGroup(288,930);
// by ID

G=gap.SmallGroup(288,930);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,56,422,100,80,9413,362,12550,1203]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b*c^-1,e*b*e=b^-1,d*c*d^-1=b^-1*c^-1,e*c*e=c^-1,e*d*e=d^5>;
// generators/relations

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