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G = C12.2C42  order 192 = 26·3

2nd non-split extension by C12 of C42 acting via C42/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C12.2C42, C6.13C4≀C2, C4⋊C4⋊4Dic3, C12.8(C4⋊C4), (C2×C12).7Q8, (C4×Dic3)⋊5C4, C4.Dic3⋊3C4, C4.2(C4×Dic3), C4.46(D6⋊C4), C3⋊2(C42⋊6C4), (C2×C4).126D12, (C2×C12).488D4, C4.2(C4⋊Dic3), (C2×C4).24Dic6, (C22×C6).43D4, C42⋊C2.2S3, (C22×C4).340D6, C12.61(C22⋊C4), C4.30(Dic3⋊C4), C22.18(D6⋊C4), C23.54(C3⋊D4), C2.1(Q8⋊3Dic3), C2.9(C6.C42), C6.8(C2.C42), C22.4(Dic3⋊C4), (C22×C12).121C22, C22.28(C6.D4), (C3×C4⋊C4)⋊6C4, (C2×C6).4(C4⋊C4), (C2×C4).68(C4×S3), (C2×C4×Dic3).1C2, (C2×C12).57(C2×C4), (C2×C4).36(C2×Dic3), (C2×C4.Dic3).8C2, (C2×C4).268(C3⋊D4), (C2×C6).90(C22⋊C4), (C3×C42⋊C2).2C2, SmallGroup(192,91)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C12.2C42
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C22×C12 — C2×C4.Dic3 — C12.2C42
Lower central
C3 — C6 — C12 — C12.2C42
Upper central
C1 — C2×C4 — C22×C4 — C42⋊C2

Generators and relations for C12.2C42
 G = < a,b,c | a12=c4=1, b4=a6, bab-1=a-1, cac-1=a7, cbc-1=a9b >

Subgroups: 248 in 110 conjugacy classes, 51 normal (39 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C42, C42⋊C2, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4.Dic3, C4×Dic3, C4×Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C42⋊6C4, C2×C4.Dic3, C2×C4×Dic3, C3×C42⋊C2, C12.2C42
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4≀C2, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C42⋊6C4, C6.C42, Q8⋊3Dic3, C12.2C42

Smallest permutation representation of C12.2C42
►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 38 23 29 7 44 17 35)(2 37 24 28 8 43 18 34)(3 48 13 27 9 42 19 33)(4 47 14 26 10 41 20 32)(5 46 15 25 11 40 21 31)(6 45 16 36 12 39 22 30)

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

48 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G4H4I4J4K···4R6A6B6C6D6E8A8B8C8D12A12B12C12D12E···12N
order122222344444444444···46666688881212121212···12
size111122211112244446···6222441212121222224···4

48 irreducible representations

dim11111112222222222224
type++++++-+-+-+
imageC1C2C2C2C4C4C4S3D4Q8D4Dic3D6Dic6C4×S3D12C3⋊D4C3⋊D4C4≀C2Q8⋊3Dic3
kernelC12.2C42C2×C4.Dic3C2×C4×Dic3C3×C42⋊C2C4.Dic3C4×Dic3C3×C4⋊C4C42⋊C2C2×C12C2×C12C22×C6C4⋊C4C22×C4C2×C4C2×C4C2×C4C2×C4C23C6C2
# reps11114441211212422284

Matrix representation of C12.2C42 ►in GL4(𝔽73) generated by

72100
72000
00460
002727
,
04600
46000
004619
001427
,
46000
04600
007271
0001
G:=sub<GL(4,GF(73))| [72,72,0,0,1,0,0,0,0,0,46,27,0,0,0,27],[0,46,0,0,46,0,0,0,0,0,46,14,0,0,19,27],[46,0,0,0,0,46,0,0,0,0,72,0,0,0,71,1] >;
 

C12.2C42 in GAP, Magma, Sage, TeX 

C_{12}._2C_4^2
 
% in TeX
 
G:=Group("C12.2C4^2");
 
// GroupNames label
 
G:=SmallGroup(192,91);
 
// by ID
 
G=gap.SmallGroup(192,91);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,184,1684,851,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=c^4=1,b^4=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;
 
// generators/relations
 

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