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G = C623C8order 288 = 25·32

1st semidirect product of C62 and C8 acting via C8/C2=C4

metabelian, soluble, monomial

Aliases: C623C8, (C2×C62).4C4, C3⋊Dic3.54D4, C326(C22⋊C8), C22⋊(C322C8), C62.11(C2×C4), (C3×C6).11M4(2), C23.2(C32⋊C4), C2.3(C62⋊C4), C2.3(C62.C4), (C3×C6).26(C2×C8), (C2×C322C8)⋊3C2, C2.5(C2×C322C8), (C2×C3⋊Dic3).17C4, C22.14(C2×C32⋊C4), (C3×C6).21(C22⋊C4), (C22×C3⋊Dic3).3C2, (C2×C3⋊Dic3).112C22, SmallGroup(288,435)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C623C8
C1C32C3×C6C3⋊Dic3C2×C3⋊Dic3C2×C322C8 — C623C8
C32C3×C6 — C623C8
C1C22C23

Generators and relations for C623C8
 G = < a,b,c | a6=b6=c8=1, ab=ba, cac-1=a-1b, cbc-1=a4b >

Subgroups: 408 in 104 conjugacy classes, 24 normal (16 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C4 [×3], C22, C22 [×2], C22 [×2], C6 [×14], C8 [×2], C2×C4 [×4], C23, C32, Dic3 [×8], C2×C6 [×14], C2×C8 [×2], C22×C4, C3×C6 [×3], C3×C6 [×2], C2×Dic3 [×12], C22×C6 [×2], C22⋊C8, C3⋊Dic3 [×2], C3⋊Dic3, C62, C62 [×2], C62 [×2], C22×Dic3 [×2], C322C8 [×2], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], C2×C62, C2×C322C8 [×2], C22×C3⋊Dic3, C623C8
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), C22⋊C8, C32⋊C4, C322C8 [×2], C2×C32⋊C4, C2×C322C8, C62.C4, C62⋊C4, C623C8

Smallest permutation representation of C623C8
On 48 points
Generators in S48
(1 29 43)(2 40 30 24 44 16)(3 45 31)(4 10 46 18 32 34)(5 25 47)(6 36 26 20 48 12)(7 41 27)(8 14 42 22 28 38)(9 17 33)(11 35 19)(13 21 37)(15 39 23)
(1 23)(2 16 44 24 30 40)(3 17)(4 34 32 18 46 10)(5 19)(6 12 48 20 26 36)(7 21)(8 38 28 22 42 14)(9 31)(11 25)(13 27)(15 29)(33 45)(35 47)(37 41)(39 43)
(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)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D4E4F6A···6N8A···8H
order122222334444446···68···8
size11112244999918184···418···18

36 irreducible representations

dim1111112244444
type+++++-+-+
imageC1C2C2C4C4C8D4M4(2)C32⋊C4C322C8C2×C32⋊C4C62.C4C62⋊C4
kernelC623C8C2×C322C8C22×C3⋊Dic3C2×C3⋊Dic3C2×C62C62C3⋊Dic3C3×C6C23C22C22C2C2
# reps1212282224244

Matrix representation of C623C8 in GL6(𝔽73)

100000
0720000
0007200
0017200
0000072
0000172
,
7200000
0720000
001000
000100
0000721
0000720
,
010000
100000
000010
000001
00712000
0018200

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,71,18,0,0,0,0,20,2,0,0,1,0,0,0,0,0,0,1,0,0] >;

C623C8 in GAP, Magma, Sage, TeX

C_6^2\rtimes_3C_8
% in TeX

G:=Group("C6^2:3C8");
// GroupNames label

G:=SmallGroup(288,435);
// by ID

G=gap.SmallGroup(288,435);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,100,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c|a^6=b^6=c^8=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=a^4*b>;
// generators/relations

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