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G = C42⋊22D6  order 192 = 26·3

20th semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42⋊22D6, C6.1262+ (1+4), (C2×Q8)⋊11D6, (C4×D12)⋊45C2, C22⋊C4⋊34D6, D6⋊D4⋊25C2, C23⋊2D6⋊24C2, Dic3⋊D4⋊42C2, C42⋊3S3⋊8C2, (C4×C12)⋊24C22, D6⋊C4⋊69C22, D6⋊3Q8⋊30C2, D6.8(C4○D4), (C2×D4).110D6, C4.4D4⋊12S3, (C6×Q8)⋊14C22, C23.9D6⋊43C2, C2.50(D4○D12), (C2×C6).222C24, C4⋊Dic3⋊41C22, Dic3⋊4D4⋊31C2, C23.14D6⋊34C2, C2.75(D4⋊6D6), C12.23D4⋊22C2, (C2×C12).631C23, Dic3⋊C4⋊36C22, C3⋊8(C22.32C24), (C4×Dic3)⋊36C22, (C6×D4).210C22, C23.8D6⋊39C2, (C22×C6).52C23, C23.54(C22×S3), (C2×D12).224C22, C22.D12⋊25C2, (C22×S3).96C23, (S3×C23).65C22, C22.243(S3×C23), (C2×Dic3).254C23, C6.D4.56C22, (C22×Dic3)⋊27C22, (S3×C2×C4)⋊52C22, C2.78(S3×C4○D4), (S3×C22⋊C4)⋊18C2, C6.189(C2×C4○D4), (C3×C4.4D4)⋊14C2, (C2×C3⋊D4)⋊24C22, (C3×C22⋊C4)⋊30C22, (C2×C4).197(C22×S3), SmallGroup(192,1237)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C42⋊22D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C42⋊22D6
Lower central
C3 — C2×C6 — C42⋊22D6

Subgroups: 752 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×10], C22, C22 [×20], S3 [×4], C6 [×3], C6 [×2], C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], Dic3 [×5], C12 [×5], D6 [×2], D6 [×12], C2×C6, C2×C6 [×6], C42, C42, C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C24, C4×S3 [×3], D12 [×3], C2×Dic3 [×5], C2×Dic3, C3⋊D4 [×5], C2×C12 [×5], C3×D4, C3×Q8, C22×S3 [×3], C22×S3 [×4], C22×C6 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C4.4D4, C42⋊2C2 [×2], C4×Dic3, Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×2], C4×C12, C3×C22⋊C4 [×4], S3×C2×C4 [×3], C2×D12 [×2], C22×Dic3, C2×C3⋊D4 [×4], C6×D4, C6×Q8, S3×C23, C22.32C24, C4×D12, C42⋊3S3, C23.8D6, S3×C22⋊C4, Dic3⋊4D4, D6⋊D4, C23.9D6, Dic3⋊D4 [×2], C22.D12, C23⋊2D6, C23.14D6, D6⋊3Q8, C12.23D4, C3×C4.4D4, C42⋊22D6

Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×2], C24, C22×S3 [×7], C2×C4○D4, 2+ (1+4) [×2], S3×C23, C22.32C24, D4⋊6D6, S3×C4○D4, D4○D12, C42⋊22D6

Generators and relations
 G = < a,b,c,d | a4=b4=c6=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽13)

120000000
012000000
00800000
00080000
0000120110
000001258
00001010
00001301
,
10000000
01000000
00130000
000120000
00001300
000081200
00000001
000000120
,
11000000
120000000
001200000
00510000
000012000
00005100
00001010
00001210012
,
11000000
012000000
00100000
008120000
000012000
000001200
00001010
00001301

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,1,1,0,0,0,0,0,12,0,3,0,0,0,0,11,5,1,0,0,0,0,0,0,8,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,3,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,5,1,12,0,0,0,0,0,1,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,1,1,0,0,0,0,0,12,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J4K4L6A6B6C6D6E12A···12F12G12H
order122222222234444444444446666612···121212
size11114466121222244446612121212222884···488

36 irreducible representations

dim1111111111111112222224444
type++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2S3D6D6D6D6C4○D42+ (1+4)D4⋊6D6S3×C4○D4D4○D12
kernelC42⋊22D6C4×D12C42⋊3S3C23.8D6S3×C22⋊C4Dic3⋊4D4D6⋊D4C23.9D6Dic3⋊D4C22.D12C23⋊2D6C23.14D6D6⋊3Q8C12.23D4C3×C4.4D4C4.4D4C42C22⋊C4C2×D4C2×Q8D6C6C2C2C2
# reps1111111121111111141142222

In GAP, Magma, Sage, TeX 

C_4^2\rtimes_{22}D_6
 
% in TeX
 
G:=Group("C4^2:22D6");
 
// GroupNames label
 
G:=SmallGroup(192,1237);
 
// by ID
 
G=gap.SmallGroup(192,1237);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,675,570,80,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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