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G = C234D12order 192 = 26·3

2nd semidirect product of C23 and D12 acting via D12/C6=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C234D12, C24.40D6, C6.42+ 1+4, D6⋊D43C2, C127D43C2, D6⋊C41C22, C22⋊C443D6, (C22×C6)⋊10D4, (C22×C4)⋊12D6, C6.8(C22×D4), (C2×D12)⋊3C22, C31(C233D4), (C2×C6).37C24, C4⋊Dic35C22, C2.8(D46D6), (S3×C23)⋊4C22, (C22×C12)⋊8C22, C2.10(C22×D12), C22.18(C2×D12), (C2×C12).130C23, C23.21D62C2, (C22×S3).9C23, (C23×C6).63C22, C22.76(S3×C23), (C22×C6).127C23, C23.158(C22×S3), (C2×Dic3).10C23, (C22×Dic3)⋊7C22, (C6×C22⋊C4)⋊15C2, (C2×C22⋊C4)⋊16S3, (C2×C6).173(C2×D4), (C22×C3⋊D4)⋊6C2, (C2×C3⋊D4)⋊36C22, (C3×C22⋊C4)⋊48C22, (C2×C4).136(C22×S3), SmallGroup(192,1052)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C234D12
C1C3C6C2×C6C22×S3S3×C23C22×C3⋊D4 — C234D12
C3C2×C6 — C234D12
C1C22C2×C22⋊C4

Generators and relations for C234D12
 G = < a,b,c,d,e | a2=b2=c2=d12=e2=1, ab=ba, dad-1=eae=ac=ca, ebe=bc=cb, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1096 in 346 conjugacy classes, 111 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×8], C22, C22 [×6], C22 [×30], S3 [×4], C6, C6 [×2], C6 [×6], C2×C4 [×4], C2×C4 [×10], D4 [×20], C23, C23 [×6], C23 [×14], Dic3 [×4], C12 [×4], D6 [×20], C2×C6, C2×C6 [×6], C2×C6 [×10], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×2], C2×D4 [×20], C24, C24 [×2], D12 [×4], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×16], C2×C12 [×4], C2×C12 [×2], C22×S3 [×4], C22×S3 [×8], C22×C6, C22×C6 [×6], C22×C6 [×2], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], C4⋊Dic3 [×4], D6⋊C4 [×8], C3×C22⋊C4 [×4], C2×D12 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×8], C2×C3⋊D4 [×8], C22×C12 [×2], S3×C23 [×2], C23×C6, C233D4, D6⋊D4 [×4], C23.21D6 [×4], C127D4 [×4], C6×C22⋊C4, C22×C3⋊D4 [×2], C234D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, 2+ 1+4 [×2], C2×D12 [×6], S3×C23, C233D4, C22×D12, D46D6 [×2], C234D12

Smallest permutation representation of C234D12
On 48 points
Generators in S48
(1 30)(2 8)(3 32)(4 10)(5 34)(6 12)(7 36)(9 26)(11 28)(13 43)(14 20)(15 45)(16 22)(17 47)(18 24)(19 37)(21 39)(23 41)(25 31)(27 33)(29 35)(38 44)(40 46)(42 48)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)
(1 36)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 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 12)(2 11)(3 10)(4 9)(5 8)(6 7)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 34)(26 33)(27 32)(28 31)(29 30)(35 36)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M 3 4A4B4C4D4E4F4G4H6A···6G6H6I6J6K12A···12H
order12222···222223444444446···6666612···12
size11112···21212121224444121212122···244444···4

42 irreducible representations

dim11111122222244
type+++++++++++++
imageC1C2C2C2C2C2S3D4D6D6D6D122+ 1+4D46D6
kernelC234D12D6⋊D4C23.21D6C127D4C6×C22⋊C4C22×C3⋊D4C2×C22⋊C4C22×C6C22⋊C4C22×C4C24C23C6C2
# reps14441214421824

Matrix representation of C234D12 in GL8(𝔽13)

10000000
01000000
001200000
000120000
000012000
00001100
000000120
00000011
,
10000000
01000000
001200000
000120000
00006030
00000603
000010070
000001007
,
10000000
01000000
00100000
00010000
000012000
000001200
000000120
000000012
,
01000000
120000000
000120000
001120000
0000121100
00000100
0000001211
00000001
,
012000000
120000000
001210000
00010000
0000121100
00000100
00004812
000009012

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,6,0,10,0,0,0,0,0,0,6,0,10,0,0,0,0,3,0,7,0,0,0,0,0,0,3,0,7],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,1],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,0,4,0,0,0,0,0,11,1,8,9,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,12] >;

C234D12 in GAP, Magma, Sage, TeX

C_2^3\rtimes_4D_{12}
% in TeX

G:=Group("C2^3:4D12");
// GroupNames label

G:=SmallGroup(192,1052);
// by ID

G=gap.SmallGroup(192,1052);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,675,570,80,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^12=e^2=1,a*b=b*a,d*a*d^-1=e*a*e=a*c=c*a,e*b*e=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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