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G = C8⋊(C32⋊C4)  order 288 = 25·32

2nd semidirect product of C8 and C32⋊C4 acting via C32⋊C4/C3⋊S3=C2

metabelian, soluble, monomial

Aliases: (C3×C24)⋊3C4, C82(C32⋊C4), C3⋊S3.5SD16, C3⋊Dic3.6Q8, C324C811C4, C324(C4.Q8), C4.8(C2×C32⋊C4), (C8×C3⋊S3).10C2, (C2×C3⋊S3).38D4, (C3×C6).11(C4⋊C4), (C3×C12).12(C2×C4), C4⋊(C32⋊C4).6C2, C2.4(C4⋊(C32⋊C4)), (C4×C3⋊S3).80C22, SmallGroup(288,416)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C8⋊(C32⋊C4)
C1C32C3×C6C2×C3⋊S3C4×C3⋊S3C4⋊(C32⋊C4) — C8⋊(C32⋊C4)
C32C3×C6C3×C12 — C8⋊(C32⋊C4)
C1C2C4C8

Generators and relations for C8⋊(C32⋊C4)
 G = < a,b,c,d | a8=b3=c3=d4=1, ab=ba, ac=ca, dad-1=a3, dcd-1=bc=cb, dbd-1=b-1c >

Subgroups: 328 in 58 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2 [×2], C3 [×2], C4, C4 [×3], C22, S3 [×4], C6 [×2], C8, C8, C2×C4 [×3], C32, Dic3 [×2], C12 [×2], D6 [×2], C4⋊C4 [×2], C2×C8, C3⋊S3 [×2], C3×C6, C3⋊C8 [×2], C24 [×2], C4×S3 [×2], C4.Q8, C3⋊Dic3, C3×C12, C32⋊C4 [×2], C2×C3⋊S3, S3×C8 [×2], C324C8, C3×C24, C4×C3⋊S3, C2×C32⋊C4 [×2], C8×C3⋊S3, C4⋊(C32⋊C4) [×2], C8⋊(C32⋊C4)
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4, Q8, C4⋊C4, SD16 [×2], C4.Q8, C32⋊C4, C2×C32⋊C4, C4⋊(C32⋊C4), C8⋊(C32⋊C4)

Character table of C8⋊(C32⋊C4)

 class 12A2B2C3A3B4A4B4C4D4E4F6A6B8A8B8C8D12A12B12C12D24A24B24C24D24E24F24G24H
 size 1199442183636363644221818444444444444
ρ1111111111111111111111111111111    trivial
ρ211111111-11-1111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111111111-11-111-1-1-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ411111111-1-1-1-1111111111111111111    linear of order 2
ρ511-1-1111-1i-i-ii11-1-1111111-1-1-1-1-1-1-1-1    linear of order 4
ρ611-1-1111-1-i-iii1111-1-1111111111111    linear of order 4
ρ711-1-1111-1ii-i-i1111-1-1111111111111    linear of order 4
ρ811-1-1111-1-iii-i11-1-1111111-1-1-1-1-1-1-1-1    linear of order 4
ρ9222222-2-20000220000-2-2-2-200000000    orthogonal lifted from D4
ρ1022-2-222-220000220000-2-2-2-200000000    symplectic lifted from Q8, Schur index 2
ρ112-22-222000000-2-2--2-2-2--20000-2-2--2--2--2--2-2-2    complex lifted from SD16
ρ122-2-2222000000-2-2-2--2-2--20000--2--2-2-2-2-2--2--2    complex lifted from SD16
ρ132-2-2222000000-2-2--2-2--2-20000-2-2--2--2--2--2-2-2    complex lifted from SD16
ρ142-22-222000000-2-2-2--2--2-20000--2--2-2-2-2-2--2--2    complex lifted from SD16
ρ154400-214000001-244001-21-2-21-2-2111-2    orthogonal lifted from C32⋊C4
ρ164400-214000001-2-4-4001-21-22-122-1-1-12    orthogonal lifted from C2×C32⋊C4
ρ1744001-2400000-214400-21-211-211-2-2-21    orthogonal lifted from C32⋊C4
ρ1844001-2400000-21-4-400-21-21-12-1-1222-1    orthogonal lifted from C2×C32⋊C4
ρ1944001-2-400000-2100002-12-1-3i03i-3i0003i    complex lifted from C4⋊(C32⋊C4)
ρ204400-21-4000001-20000-12-1203i003i-3i-3i0    complex lifted from C4⋊(C32⋊C4)
ρ214400-21-4000001-20000-12-120-3i00-3i3i3i0    complex lifted from C4⋊(C32⋊C4)
ρ2244001-2-400000-2100002-12-13i0-3i3i000-3i    complex lifted from C4⋊(C32⋊C4)
ρ234-4001-20000002-1-2-22-2000-3i03i838--287+2ζ858785-2-2--283+2ζ8    complex faithful
ρ244-4001-20000002-12-2-2-2000-3i03i8785-283+2ζ8838--2--2-287+2ζ85    complex faithful
ρ254-4001-20000002-12-2-2-20003i0-3i87+2ζ85-283883+2ζ8--2--2-28785    complex faithful
ρ264-400-21000000-12-2-22-200-3i03i0--283+2ζ8-2-287+2ζ858785838--2    complex faithful
ρ274-4001-20000002-1-2-22-20003i0-3i83+2ζ8--2878587+2ζ85-2-2--2838    complex faithful
ρ284-400-21000000-12-2-22-2003i0-3i0--2838-2-2878587+2ζ8583+2ζ8--2    complex faithful
ρ294-400-21000000-122-2-2-2003i0-3i0-28785--2--283883+2ζ887+2ζ85-2    complex faithful
ρ304-400-21000000-122-2-2-200-3i03i0-287+2ζ85--2--283+2ζ88388785-2    complex faithful

Smallest permutation representation of C8⋊(C32⋊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)
(9 21 48)(10 22 41)(11 23 42)(12 24 43)(13 17 44)(14 18 45)(15 19 46)(16 20 47)
(1 34 29)(2 35 30)(3 36 31)(4 37 32)(5 38 25)(6 39 26)(7 40 27)(8 33 28)(9 21 48)(10 22 41)(11 23 42)(12 24 43)(13 17 44)(14 18 45)(15 19 46)(16 20 47)
(1 15)(2 10)(3 13)(4 16)(5 11)(6 14)(7 9)(8 12)(17 31 44 36)(18 26 45 39)(19 29 46 34)(20 32 47 37)(21 27 48 40)(22 30 41 35)(23 25 42 38)(24 28 43 33)

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

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

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

Matrix representation of C8⋊(C32⋊C4) in GL6(𝔽73)

6760000
67670000
0027000
0002700
0000460
0000046
,
100000
010000
001000
000100
0000721
0000720
,
100000
010000
0007200
0017200
0000721
0000720
,
8640000
64650000
000010
000001
0017200
0007200

G:=sub<GL(6,GF(73))| [67,67,0,0,0,0,6,67,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[8,64,0,0,0,0,64,65,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,1,0,0,0,0,0,0,1,0,0] >;

C8⋊(C32⋊C4) in GAP, Magma, Sage, TeX

C_8\rtimes (C_3^2\rtimes C_4)
% in TeX

G:=Group("C8:(C3^2:C4)");
// GroupNames label

G:=SmallGroup(288,416);
// by ID

G=gap.SmallGroup(288,416);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,64,675,80,9413,691,12550,2372]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^3=c^3=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^3,d*c*d^-1=b*c=c*b,d*b*d^-1=b^-1*c>;
// generators/relations

Export

Character table of C8⋊(C32⋊C4) in TeX

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