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G = C325(C4⋊C8)  order 288 = 25·32

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

metabelian, soluble, monomial

Aliases: C325(C4⋊C8), (C6×C12).8C4, C3⋊Dic37C8, C3⋊Dic3.9Q8, C62.7(C2×C4), C3⋊Dic3.38D4, (C3×C6).10M4(2), C2.2(C62.C4), (C3×C6).11(C2×C8), (C3×C6).16(C4⋊C4), (C2×C4).4(C32⋊C4), C2.5(C3⋊S33C8), C2.2(C4⋊(C32⋊C4)), (C2×C3⋊Dic3).16C4, (C4×C3⋊Dic3).17C2, (C2×C322C8).3C2, C22.12(C2×C32⋊C4), (C2×C3⋊Dic3).111C22, SmallGroup(288,427)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C325(C4⋊C8)
C1C32C3×C6C3⋊Dic3C2×C3⋊Dic3C2×C322C8 — C325(C4⋊C8)
C32C3×C6 — C325(C4⋊C8)
C1C22C2×C4

Generators and relations for C325(C4⋊C8)
 G = < a,b,c,d | a3=b3=c4=d8=1, dbd-1=ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dcd-1=c-1 >

Subgroups: 272 in 68 conjugacy classes, 22 normal (18 characteristic)
C1, C2 [×3], C3 [×2], C4 [×5], C22, C6 [×6], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×8], C12 [×4], C2×C6 [×2], C42, C2×C8 [×2], C3×C6 [×3], C2×Dic3 [×4], C2×C12 [×2], C4⋊C8, C3⋊Dic3 [×2], C3⋊Dic3 [×2], C3×C12, C62, C4×Dic3 [×2], C322C8 [×2], C2×C3⋊Dic3 [×2], C6×C12, C4×C3⋊Dic3, C2×C322C8 [×2], C325(C4⋊C8)
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4, Q8, C4⋊C4, C2×C8, M4(2), C4⋊C8, C32⋊C4, C2×C32⋊C4, C3⋊S33C8, C4⋊(C32⋊C4), C62.C4, C325(C4⋊C8)

Smallest permutation representation of C325(C4⋊C8)
On 96 points
Generators in S96
(2 68 44)(4 46 70)(6 72 48)(8 42 66)(10 28 34)(12 36 30)(14 32 38)(16 40 26)(18 96 86)(20 88 90)(22 92 82)(24 84 94)(50 74 60)(52 62 76)(54 78 64)(56 58 80)
(1 67 43)(2 68 44)(3 45 69)(4 46 70)(5 71 47)(6 72 48)(7 41 65)(8 42 66)(9 27 33)(10 28 34)(11 35 29)(12 36 30)(13 31 37)(14 32 38)(15 39 25)(16 40 26)(17 95 85)(18 96 86)(19 87 89)(20 88 90)(21 91 81)(22 92 82)(23 83 93)(24 84 94)(49 73 59)(50 74 60)(51 61 75)(52 62 76)(53 77 63)(54 78 64)(55 57 79)(56 58 80)
(1 75 17 13)(2 14 18 76)(3 77 19 15)(4 16 20 78)(5 79 21 9)(6 10 22 80)(7 73 23 11)(8 12 24 74)(25 45 53 87)(26 88 54 46)(27 47 55 81)(28 82 56 48)(29 41 49 83)(30 84 50 42)(31 43 51 85)(32 86 52 44)(33 71 57 91)(34 92 58 72)(35 65 59 93)(36 94 60 66)(37 67 61 95)(38 96 62 68)(39 69 63 89)(40 90 64 70)
(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)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)

G:=sub<Sym(96)| (2,68,44)(4,46,70)(6,72,48)(8,42,66)(10,28,34)(12,36,30)(14,32,38)(16,40,26)(18,96,86)(20,88,90)(22,92,82)(24,84,94)(50,74,60)(52,62,76)(54,78,64)(56,58,80), (1,67,43)(2,68,44)(3,45,69)(4,46,70)(5,71,47)(6,72,48)(7,41,65)(8,42,66)(9,27,33)(10,28,34)(11,35,29)(12,36,30)(13,31,37)(14,32,38)(15,39,25)(16,40,26)(17,95,85)(18,96,86)(19,87,89)(20,88,90)(21,91,81)(22,92,82)(23,83,93)(24,84,94)(49,73,59)(50,74,60)(51,61,75)(52,62,76)(53,77,63)(54,78,64)(55,57,79)(56,58,80), (1,75,17,13)(2,14,18,76)(3,77,19,15)(4,16,20,78)(5,79,21,9)(6,10,22,80)(7,73,23,11)(8,12,24,74)(25,45,53,87)(26,88,54,46)(27,47,55,81)(28,82,56,48)(29,41,49,83)(30,84,50,42)(31,43,51,85)(32,86,52,44)(33,71,57,91)(34,92,58,72)(35,65,59,93)(36,94,60,66)(37,67,61,95)(38,96,62,68)(39,69,63,89)(40,90,64,70), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)>;

G:=Group( (2,68,44)(4,46,70)(6,72,48)(8,42,66)(10,28,34)(12,36,30)(14,32,38)(16,40,26)(18,96,86)(20,88,90)(22,92,82)(24,84,94)(50,74,60)(52,62,76)(54,78,64)(56,58,80), (1,67,43)(2,68,44)(3,45,69)(4,46,70)(5,71,47)(6,72,48)(7,41,65)(8,42,66)(9,27,33)(10,28,34)(11,35,29)(12,36,30)(13,31,37)(14,32,38)(15,39,25)(16,40,26)(17,95,85)(18,96,86)(19,87,89)(20,88,90)(21,91,81)(22,92,82)(23,83,93)(24,84,94)(49,73,59)(50,74,60)(51,61,75)(52,62,76)(53,77,63)(54,78,64)(55,57,79)(56,58,80), (1,75,17,13)(2,14,18,76)(3,77,19,15)(4,16,20,78)(5,79,21,9)(6,10,22,80)(7,73,23,11)(8,12,24,74)(25,45,53,87)(26,88,54,46)(27,47,55,81)(28,82,56,48)(29,41,49,83)(30,84,50,42)(31,43,51,85)(32,86,52,44)(33,71,57,91)(34,92,58,72)(35,65,59,93)(36,94,60,66)(37,67,61,95)(38,96,62,68)(39,69,63,89)(40,90,64,70), (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)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96) );

G=PermutationGroup([(2,68,44),(4,46,70),(6,72,48),(8,42,66),(10,28,34),(12,36,30),(14,32,38),(16,40,26),(18,96,86),(20,88,90),(22,92,82),(24,84,94),(50,74,60),(52,62,76),(54,78,64),(56,58,80)], [(1,67,43),(2,68,44),(3,45,69),(4,46,70),(5,71,47),(6,72,48),(7,41,65),(8,42,66),(9,27,33),(10,28,34),(11,35,29),(12,36,30),(13,31,37),(14,32,38),(15,39,25),(16,40,26),(17,95,85),(18,96,86),(19,87,89),(20,88,90),(21,91,81),(22,92,82),(23,83,93),(24,84,94),(49,73,59),(50,74,60),(51,61,75),(52,62,76),(53,77,63),(54,78,64),(55,57,79),(56,58,80)], [(1,75,17,13),(2,14,18,76),(3,77,19,15),(4,16,20,78),(5,79,21,9),(6,10,22,80),(7,73,23,11),(8,12,24,74),(25,45,53,87),(26,88,54,46),(27,47,55,81),(28,82,56,48),(29,41,49,83),(30,84,50,42),(31,43,51,85),(32,86,52,44),(33,71,57,91),(34,92,58,72),(35,65,59,93),(36,94,60,66),(37,67,61,95),(38,96,62,68),(39,69,63,89),(40,90,64,70)], [(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),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96)])

36 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H6A···6F8A···8H12A···12H
order122233444444446···68···812···12
size11114422999918184···418···184···4

36 irreducible representations

dim11111122244444
type++++-++-
imageC1C2C2C4C4C8D4Q8M4(2)C32⋊C4C2×C32⋊C4C3⋊S33C8C4⋊(C32⋊C4)C62.C4
kernelC325(C4⋊C8)C4×C3⋊Dic3C2×C322C8C2×C3⋊Dic3C6×C12C3⋊Dic3C3⋊Dic3C3⋊Dic3C3×C6C2×C4C22C2C2C2
# reps11222811222444

Matrix representation of C325(C4⋊C8) in GL6(𝔽73)

100000
010000
001000
000100
0000072
0000172
,
100000
010000
0072100
0072000
0000072
0000172
,
4670000
0270000
00422800
00703100
00004228
00007031
,
64150000
1490000
000010
000001
00344700
0083900

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[46,0,0,0,0,0,7,27,0,0,0,0,0,0,42,70,0,0,0,0,28,31,0,0,0,0,0,0,42,70,0,0,0,0,28,31],[64,14,0,0,0,0,15,9,0,0,0,0,0,0,0,0,34,8,0,0,0,0,47,39,0,0,1,0,0,0,0,0,0,1,0,0] >;

C325(C4⋊C8) in GAP, Magma, Sage, TeX

C_3^2\rtimes_5(C_4\rtimes C_8)
% in TeX

G:=Group("C3^2:5(C4:C8)");
// GroupNames label

G:=SmallGroup(288,427);
// by ID

G=gap.SmallGroup(288,427);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,176,100,9413,691,12550,2372]);
// Polycyclic

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

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