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G = C624D4order 288 = 25·32

1st semidirect product of C62 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C624D4, C62.118C23, (S3×C6)⋊15D4, C23.26S32, D67(C3⋊D4), (S3×C23)⋊4S3, (C2×Dic3)⋊3D6, C6.170(S3×D4), C324C22≀C2, (C22×S3)⋊3D6, C35(C232D6), D6⋊Dic335C2, C625C410C2, C32(C244S3), (C22×C6).118D6, C223(D6⋊S3), (C6×Dic3)⋊15C22, (C2×C62).37C22, (C2×C3⋊D4)⋊4S3, (S3×C22×C6)⋊1C2, (S3×C2×C6)⋊2C22, (C6×C3⋊D4)⋊10C2, C2.42(S3×C3⋊D4), C6.84(C2×C3⋊D4), (C2×D6⋊S3)⋊8C2, (C2×C6)⋊14(C3⋊D4), C22.141(C2×S32), (C3×C6).164(C2×D4), C2.16(C2×D6⋊S3), (C2×C3⋊Dic3)⋊4C22, (C2×C6).137(C22×S3), SmallGroup(288,624)

Series: Derived Chief Lower central Upper central

C1C62 — C624D4
C1C3C32C3×C6C62S3×C2×C6C2×D6⋊S3 — C624D4
C32C62 — C624D4
C1C22C23

Generators and relations for C624D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1018 in 277 conjugacy classes, 60 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×7], C3 [×2], C3, C4 [×3], C22, C22 [×2], C22 [×21], S3 [×5], C6 [×2], C6 [×4], C6 [×14], C2×C4 [×3], D4 [×6], C23, C23 [×9], C32, Dic3 [×7], C12, D6 [×4], D6 [×15], C2×C6 [×2], C2×C6 [×4], C2×C6 [×28], C22⋊C4 [×3], C2×D4 [×3], C24, C3×S3 [×5], C3×C6, C3×C6 [×2], C3×C6 [×2], C2×Dic3, C2×Dic3 [×6], C3⋊D4 [×10], C2×C12, C3×D4 [×2], C22×S3, C22×S3 [×2], C22×S3 [×6], C22×C6 [×2], C22×C6 [×10], C22≀C2, C3×Dic3, C3⋊Dic3 [×2], S3×C6 [×4], S3×C6 [×15], C62, C62 [×2], C62 [×2], D6⋊C4 [×2], C6.D4 [×5], C2×C3⋊D4, C2×C3⋊D4 [×4], C6×D4, S3×C23, C23×C6, D6⋊S3 [×4], C6×Dic3, C3×C3⋊D4 [×2], C2×C3⋊Dic3 [×2], S3×C2×C6, S3×C2×C6 [×2], S3×C2×C6 [×6], C2×C62, C232D6, C244S3, D6⋊Dic3 [×2], C625C4, C2×D6⋊S3 [×2], C6×C3⋊D4, S3×C22×C6, C624D4
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×6], C23, D6 [×6], C2×D4 [×3], C3⋊D4 [×8], C22×S3 [×2], C22≀C2, S32, S3×D4 [×2], C2×C3⋊D4 [×4], D6⋊S3 [×2], C2×S32, C232D6, C244S3, C2×D6⋊S3, S3×C3⋊D4 [×2], C624D4

Smallest permutation representation of C624D4
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)
(1 15 3 14 2 13)(4 20 5 21 6 19)(7 12 8 10 9 11)(16 24 18 23 17 22)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 42 41 40 39 38)(43 48 47 46 45 44)
(1 30 24 33)(2 26 22 35)(3 28 23 31)(4 44 10 41)(5 46 11 37)(6 48 12 39)(7 38 21 47)(8 40 19 43)(9 42 20 45)(13 25 16 34)(14 27 17 36)(15 29 18 32)
(1 20)(2 21)(3 19)(4 15)(5 13)(6 14)(7 22)(8 23)(9 24)(10 18)(11 16)(12 17)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C6A···6J6K···6S6T···6AA6AB6AC12A12B
order122222222223334446···66···66···6661212
size1111226666122241236362···24···46···612121212

48 irreducible representations

dim11111122222222244444
type+++++++++++++++-+
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C3⋊D4C3⋊D4S32S3×D4D6⋊S3C2×S32S3×C3⋊D4
kernelC624D4D6⋊Dic3C625C4C2×D6⋊S3C6×C3⋊D4S3×C22×C6C2×C3⋊D4S3×C23S3×C6C62C2×Dic3C22×S3C22×C6D6C2×C6C23C6C22C22C2
# reps12121111421328812214

Matrix representation of C624D4 in GL8(𝔽13)

120000000
012000000
00100000
000120000
00001000
000031200
000000012
00000011
,
01000000
1212000000
001200000
000120000
000012000
000001200
00000010
00000001
,
10000000
1212000000
00010000
001200000
000012500
00000100
000000120
00000011
,
10000000
1212000000
00100000
000120000
000012000
000001200
00000010
00000001

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1],[0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1],[1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;

C624D4 in GAP, Magma, Sage, TeX

C_6^2\rtimes_4D_4
% in TeX

G:=Group("C6^2:4D4");
// GroupNames label

G:=SmallGroup(288,624);
// by ID

G=gap.SmallGroup(288,624);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,219,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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