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G = C4⋊C436D6order 192 = 26·3

2nd semidirect product of C4⋊C4 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C436D6, (C2×D12)⋊11C4, C4.62(C2×D12), (C2×C4).46D12, C42⋊C21S3, C4.23(D6⋊C4), D12.25(C2×C4), C6.D827C2, C12.142(C2×D4), (C2×C12).472D4, (C22×C6).74D4, C2.2(D4⋊D6), C12.64(C22×C4), (C22×C4).126D6, C6.105(C8⋊C22), C12.47(C22⋊C4), (C2×C12).328C23, C22.23(D6⋊C4), (C22×D12).12C2, C23.61(C3⋊D4), C32(C23.37D4), (C2×D12).234C22, (C22×C12).150C22, C4.51(S3×C2×C4), (C2×C3⋊C8)⋊4C22, (C2×C4).44(C4×S3), C2.17(C2×D6⋊C4), (C3×C4⋊C4)⋊41C22, (C2×C12).91(C2×C4), (C2×C6).457(C2×D4), C6.44(C2×C22⋊C4), (C2×C4.Dic3)⋊9C2, (C3×C42⋊C2)⋊1C2, C22.72(C2×C3⋊D4), (C2×C4).241(C3⋊D4), (C2×C6).14(C22⋊C4), (C2×C4).428(C22×S3), SmallGroup(192,560)

Series: Derived Chief Lower central Upper central

C1C12 — C4⋊C436D6
C1C3C6C2×C6C2×C12C2×D12C22×D12 — C4⋊C436D6
C3C6C12 — C4⋊C436D6
C1C22C22×C4C42⋊C2

Generators and relations for C4⋊C436D6
 G = < a,b,c,d | a4=b4=c6=d2=1, bab-1=dad=a-1, ac=ca, cbc-1=a2b, dbd=ab-1, dcd=c-1 >

Subgroups: 680 in 190 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×18], S3 [×4], C6, C6 [×2], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], D4 [×10], C23, C23 [×10], C12 [×2], C12 [×2], C12 [×2], D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C2×D4 [×9], C24, C3⋊C8 [×2], D12 [×4], D12 [×6], C2×C12 [×2], C2×C12 [×4], C2×C12 [×2], C22×S3 [×10], C22×C6, D4⋊C4 [×4], C42⋊C2, C2×M4(2), C22×D4, C2×C3⋊C8 [×2], C4.Dic3 [×2], C4×C12, C3×C22⋊C4, C3×C4⋊C4 [×2], C2×D12 [×6], C2×D12 [×3], C22×C12, S3×C23, C23.37D4, C6.D8 [×4], C2×C4.Dic3, C3×C42⋊C2, C22×D12, C4⋊C436D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C8⋊C22 [×2], D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C23.37D4, C2×D6⋊C4, D4⋊D6 [×2], C4⋊C436D6

Smallest permutation representation of C4⋊C436D6
On 48 points
Generators in S48
(1 9 19 29)(2 10 20 30)(3 11 21 25)(4 12 22 26)(5 7 23 27)(6 8 24 28)(13 41 33 44)(14 42 34 45)(15 37 35 46)(16 38 36 47)(17 39 31 48)(18 40 32 43)
(1 43 22 46)(2 41 23 38)(3 45 24 48)(4 37 19 40)(5 47 20 44)(6 39 21 42)(7 36 30 33)(8 17 25 14)(9 32 26 35)(10 13 27 16)(11 34 28 31)(12 15 29 18)
(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 3)(4 6)(7 27)(8 26)(9 25)(10 30)(11 29)(12 28)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 21)(22 24)(31 43)(32 48)(33 47)(34 46)(35 45)(36 44)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D12A12B12C12D12E···12N
order12222222223444444446666688881212121212···12
size11112212121212222224444222441212121222224···4

42 irreducible representations

dim11111122222222244
type+++++++++++++
imageC1C2C2C2C2C4S3D4D4D6D6C4×S3D12C3⋊D4C3⋊D4C8⋊C22D4⋊D6
kernelC4⋊C436D6C6.D8C2×C4.Dic3C3×C42⋊C2C22×D12C2×D12C42⋊C2C2×C12C22×C6C4⋊C4C22×C4C2×C4C2×C4C2×C4C23C6C2
# reps14111813121442224

Matrix representation of C4⋊C436D6 in GL6(𝔽73)

7200000
0720000
0075900
00146600
00006614
0000597
,
4600000
0460000
0000720
0000072
001000
000100
,
72720000
100000
0007200
0017200
000001
0000721
,
110000
0720000
0072100
000100
000077
00001466

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,0,0,0,0,66,59,0,0,0,0,14,7],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,72,0,0,0,0,0,0,72,0,0],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,72,0,0,0,0,0,1,1,0,0,0,0,0,0,7,14,0,0,0,0,7,66] >;

C4⋊C436D6 in GAP, Magma, Sage, TeX

C_4\rtimes C_4\rtimes_{36}D_6
% in TeX

G:=Group("C4:C4:36D6");
// GroupNames label

G:=SmallGroup(192,560);
// by ID

G=gap.SmallGroup(192,560);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,1684,438,102,6278]);
// Polycyclic

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

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