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G = C623C12order 432 = 24·33

2nd semidirect product of C62 and C12 acting via C12/C2=C6

metabelian, supersoluble, monomial

Aliases: C623C12, C623Dic3, C62.26D6, C625C4⋊C3, (C2×C62).2S3, (C2×C62).4C6, C62.8(C2×C6), He37(C22⋊C4), (C2×He3).31D4, (C22×He3)⋊4C4, C6.19(C6×Dic3), C2.3(He36D4), (C23×He3).2C2, C223(C32⋊C12), C23.3(C32⋊C6), C323(C6.D4), (C22×He3).20C22, (C3×C6).9(C2×C12), (C2×C6).47(S3×C6), (C2×C3⋊Dic3)⋊2C6, (C3×C6).16(C3×D4), C6.31(C3×C3⋊D4), (C2×C32⋊C12)⋊4C2, C2.5(C2×C32⋊C12), C323(C3×C22⋊C4), (C3×C6).31(C3⋊D4), (C2×He3).30(C2×C4), (C22×C6).25(C3×S3), (C2×C6).19(C3×Dic3), (C3×C6).14(C2×Dic3), C3.2(C3×C6.D4), C22.7(C2×C32⋊C6), SmallGroup(432,166)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C623C12
C1C3C32C3×C6C62C22×He3C2×C32⋊C12 — C623C12
C32C3×C6 — C623C12
C1C22C23

Generators and relations for C623C12
 G = < a,b,c | a6=b6=c12=1, ab=ba, cac-1=a-1b, cbc-1=b-1 >

Subgroups: 549 in 151 conjugacy classes, 46 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C3 [×3], C4 [×2], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×19], C2×C4 [×2], C23, C32 [×2], C32, Dic3 [×4], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×19], C22⋊C4, C3×C6 [×2], C3×C6 [×4], C3×C6 [×11], C2×Dic3 [×4], C2×C12 [×2], C22×C6, C22×C6 [×3], He3, C3×Dic3 [×2], C3⋊Dic3 [×2], C62 [×2], C62 [×4], C62 [×11], C6.D4 [×2], C3×C22⋊C4, C2×He3, C2×He3 [×2], C2×He3 [×2], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C2×C62 [×2], C2×C62, C32⋊C12 [×2], C22×He3, C22×He3 [×2], C22×He3 [×2], C3×C6.D4, C625C4, C2×C32⋊C12 [×2], C23×He3, C623C12
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4 [×2], Dic3 [×2], C12 [×2], D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4 [×2], C2×C12, C3×D4 [×2], C3×Dic3 [×2], S3×C6, C6.D4, C3×C22⋊C4, C32⋊C6, C6×Dic3, C3×C3⋊D4 [×2], C32⋊C12 [×2], C2×C32⋊C6, C3×C6.D4, C2×C32⋊C12, He36D4 [×2], C623C12

Smallest permutation representation of C623C12
On 72 points
Generators in S72
(1 11 59 7 5 53)(2 48 56 40 10 15)(3 70 68 9 64 62)(4 21 8 42 50 46)(6 27 67 44 71 35)(12 33 61 38 65 29)(13 69 17 54 25 58)(14 28 22 20 34 16)(18 45 43 24 39 37)(19 63 23 60 31 52)(26 36 47 32 30 41)(49 66 57 55 72 51)
(1 32 51 45 70 16)(2 17 71 46 52 33)(3 34 53 47 72 18)(4 19 61 48 54 35)(5 36 55 37 62 20)(6 21 63 38 56 25)(7 26 57 39 64 22)(8 23 65 40 58 27)(9 28 59 41 66 24)(10 13 67 42 60 29)(11 30 49 43 68 14)(12 15 69 44 50 31)
(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)

G:=sub<Sym(72)| (1,11,59,7,5,53)(2,48,56,40,10,15)(3,70,68,9,64,62)(4,21,8,42,50,46)(6,27,67,44,71,35)(12,33,61,38,65,29)(13,69,17,54,25,58)(14,28,22,20,34,16)(18,45,43,24,39,37)(19,63,23,60,31,52)(26,36,47,32,30,41)(49,66,57,55,72,51), (1,32,51,45,70,16)(2,17,71,46,52,33)(3,34,53,47,72,18)(4,19,61,48,54,35)(5,36,55,37,62,20)(6,21,63,38,56,25)(7,26,57,39,64,22)(8,23,65,40,58,27)(9,28,59,41,66,24)(10,13,67,42,60,29)(11,30,49,43,68,14)(12,15,69,44,50,31), (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)>;

G:=Group( (1,11,59,7,5,53)(2,48,56,40,10,15)(3,70,68,9,64,62)(4,21,8,42,50,46)(6,27,67,44,71,35)(12,33,61,38,65,29)(13,69,17,54,25,58)(14,28,22,20,34,16)(18,45,43,24,39,37)(19,63,23,60,31,52)(26,36,47,32,30,41)(49,66,57,55,72,51), (1,32,51,45,70,16)(2,17,71,46,52,33)(3,34,53,47,72,18)(4,19,61,48,54,35)(5,36,55,37,62,20)(6,21,63,38,56,25)(7,26,57,39,64,22)(8,23,65,40,58,27)(9,28,59,41,66,24)(10,13,67,42,60,29)(11,30,49,43,68,14)(12,15,69,44,50,31), (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) );

G=PermutationGroup([(1,11,59,7,5,53),(2,48,56,40,10,15),(3,70,68,9,64,62),(4,21,8,42,50,46),(6,27,67,44,71,35),(12,33,61,38,65,29),(13,69,17,54,25,58),(14,28,22,20,34,16),(18,45,43,24,39,37),(19,63,23,60,31,52),(26,36,47,32,30,41),(49,66,57,55,72,51)], [(1,32,51,45,70,16),(2,17,71,46,52,33),(3,34,53,47,72,18),(4,19,61,48,54,35),(5,36,55,37,62,20),(6,21,63,38,56,25),(7,26,57,39,64,22),(8,23,65,40,58,27),(9,28,59,41,66,24),(10,13,67,42,60,29),(11,30,49,43,68,14),(12,15,69,44,50,31)], [(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)])

62 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E3F4A4B4C4D6A···6G6H···6M6N···6AL12A···12H
order12222233333344446···66···66···612···12
size111122233666181818182···23···36···618···18

62 irreducible representations

dim1111111122222222226666
type+++++-++-+
imageC1C2C2C3C4C6C6C12S3D4Dic3D6C3×S3C3⋊D4C3×D4C3×Dic3S3×C6C3×C3⋊D4C32⋊C6C32⋊C12C2×C32⋊C6He36D4
kernelC623C12C2×C32⋊C12C23×He3C625C4C22×He3C2×C3⋊Dic3C2×C62C62C2×C62C2×He3C62C62C22×C6C3×C6C3×C6C2×C6C2×C6C6C23C22C22C2
# reps1212442812212444281214

Matrix representation of C623C12 in GL8(𝔽13)

90000000
03000000
000012000
001200000
000120000
00000010
00000001
00000100
,
10000000
01000000
001000000
000100000
000010000
00000400
00000040
00000004
,
01000000
120000000
00000009
00000100
00000030
00004000
001200000
000100000

G:=sub<GL(8,GF(13))| [9,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,10,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0] >;

C623C12 in GAP, Magma, Sage, TeX

C_6^2\rtimes_3C_{12}
% in TeX

G:=Group("C6^2:3C12");
// GroupNames label

G:=SmallGroup(432,166);
// by ID

G=gap.SmallGroup(432,166);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,4037,2035,14118]);
// Polycyclic

G:=Group<a,b,c|a^6=b^6=c^12=1,a*b=b*a,c*a*c^-1=a^-1*b,c*b*c^-1=b^-1>;
// generators/relations

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