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G = C23⋊4D12  order 192 = 26·3

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23⋊4D12, C24.40D6, C6.42+ (1+4), D6⋊D4⋊3C2, C12⋊7D4⋊3C2, D6⋊C4⋊1C22, C22⋊C4⋊43D6, (C22×C6)⋊10D4, (C22×C4)⋊12D6, C6.8(C22×D4), (C2×D12)⋊3C22, C3⋊1(C23⋊3D4), (C2×C6).37C24, C4⋊Dic3⋊5C22, C2.8(D4⋊6D6), (S3×C23)⋊4C22, (C22×C12)⋊8C22, C22.18(C2×D12), C2.10(C22×D12), (C2×C12).130C23, C22.D12⋊2C2, (C22×S3).9C23, C22.76(S3×C23), (C23×C6).63C22, (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

Derived series
C1 — C2×C6 — C23⋊4D12
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×C3⋊D4 — C23⋊4D12
Lower central
C3 — C2×C6 — C23⋊4D12

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, C23⋊3D4, D6⋊D4 [×4], C22.D12 [×4], C12⋊7D4 [×4], C6×C22⋊C4, C22×C3⋊D4 [×2], C23⋊4D12

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, C23⋊3D4, C22×D12, D4⋊6D6 [×2], C23⋊4D12

Generators and relations
 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 >

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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+4)D4⋊6D6
kernelC23⋊4D12D6⋊D4C22.D12C12⋊7D4C6×C22⋊C4C22×C3⋊D4C2×C22⋊C4C22×C6C22⋊C4C22×C4C24C23C6C2
# reps14441214421824

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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