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G = C24.44D6  order 192 = 26·3

33rd non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.44D6, C6.272+ (1+4), C3⋊D4⋊5D4, (C2×D4)⋊19D6, C22⋊C4⋊6D6, C22≀C2⋊5S3, C23⋊2D6⋊5C2, C3⋊4(D4⋊5D4), D6.16(C2×D4), (C6×D4)⋊8C22, D6⋊3D4⋊13C2, D6⋊C4⋊12C22, (D4×Dic3)⋊12C2, C6.57(C22×D4), C22.11(S3×D4), C23.14D6⋊3C2, Dic3⋊4D4⋊3C2, C23.9D6⋊13C2, (C2×C12).29C23, (C2×C6).135C24, C4⋊Dic3⋊26C22, Dic3.19(C2×D4), (C22×C6).9C23, C2.29(D4⋊6D6), C22⋊5(D4⋊2S3), Dic3⋊C4⋊10C22, (C4×Dic3)⋊15C22, (C2×Dic6)⋊20C22, (C23×C6).68C22, C23.17(C22×S3), C23.11D6⋊12C2, Dic3.D4⋊13C2, C6.D4⋊50C22, C22.D12⋊10C2, (S3×C23).43C22, (C22×S3).54C23, C22.156(S3×C23), (C2×Dic3).222C23, (C22×Dic3)⋊14C22, C2.30(C2×S3×D4), (S3×C2×C4)⋊8C22, (S3×C22⋊C4)⋊3C2, C6.77(C2×C4○D4), (C2×C6).54(C2×D4), (C3×C22≀C2)⋊6C2, (C2×D4⋊2S3)⋊6C2, (C2×C6)⋊10(C4○D4), (C2×C3⋊D4)⋊8C22, (C22×C3⋊D4)⋊9C2, C2.28(C2×D4⋊2S3), (C3×C22⋊C4)⋊6C22, (C2×C4).29(C22×S3), (C2×C6.D4)⋊20C2, SmallGroup(192,1150)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C24.44D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22⋊C4 — C24.44D6
Lower central
C3 — C2×C6 — C24.44D6

Subgroups: 928 in 334 conjugacy classes, 107 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C3, C4 [×10], C22, C22 [×4], C22 [×25], S3 [×3], C6 [×3], C6 [×6], C2×C4 [×3], C2×C4 [×16], D4 [×18], Q8 [×2], C23 [×4], C23 [×12], Dic3 [×2], Dic3 [×5], C12 [×3], D6 [×2], D6 [×9], C2×C6, C2×C6 [×4], C2×C6 [×14], C42, C22⋊C4 [×3], C22⋊C4 [×9], C4⋊C4 [×4], C22×C4 [×6], C2×D4 [×3], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24, C24, Dic6 [×2], C4×S3 [×3], C2×Dic3 [×6], C2×Dic3 [×7], C3⋊D4 [×4], C3⋊D4 [×9], C2×C12 [×3], C3×D4 [×5], C22×S3 [×2], C22×S3 [×5], C22×C6 [×4], C22×C6 [×5], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2, C22≀C2, C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×5], C3×C22⋊C4 [×3], C2×Dic6, S3×C2×C4 [×2], D4⋊2S3 [×4], C22×Dic3 [×4], C2×C3⋊D4 [×6], C2×C3⋊D4 [×4], C6×D4 [×3], S3×C23, C23×C6, D4⋊5D4, Dic3.D4, S3×C22⋊C4, Dic3⋊4D4, C23.9D6, C23.11D6, C22.D12, D4×Dic3, C23⋊2D6, D6⋊3D4, C23.14D6 [×2], C2×C6.D4, C3×C22≀C2, C2×D4⋊2S3, C22×C3⋊D4, C24.44D6

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×D4, C2×C4○D4, 2+ (1+4), S3×D4 [×2], D4⋊2S3 [×2], S3×C23, D4⋊5D4, C2×S3×D4, C2×D4⋊2S3, D4⋊6D6, C24.44D6

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

1200000
010000
0012000
0001200
000010
000001
,
1200000
010000
0012200
000100
000010
000001
,
1200000
0120000
001000
000100
000010
000001
,
1200000
0120000
0012000
0001200
000010
000001
,
010000
1200000
008000
008500
000001
0000121
,
0120000
100000
005000
000500
0000121
000001

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,8,0,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,1,1] >;
 

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L 3 4A4B4C4D···4I4J4K4L6A6B6C6D···6I6J12A12B12C
order122222222222234444···44446666···66121212
size1111222244661224446···61212122224···48888

39 irreducible representations

dim1111111111111112222224444
type++++++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2S3D4D6D6D6C4○D42+ (1+4)S3×D4D4⋊2S3D4⋊6D6
kernelC24.44D6Dic3.D4S3×C22⋊C4Dic3⋊4D4C23.9D6C23.11D6C22.D12D4×Dic3C23⋊2D6D6⋊3D4C23.14D6C2×C6.D4C3×C22≀C2C2×D4⋊2S3C22×C3⋊D4C22≀C2C3⋊D4C22⋊C4C2×D4C24C2×C6C6C22C22C2
# reps1111111111211111433141222

In GAP, Magma, Sage, TeX 

C_2^4._{44}D_6
 
% in TeX
 
G:=Group("C2^4.44D6");
 
// GroupNames label
 
G:=SmallGroup(192,1150);
 
// by ID
 
G=gap.SmallGroup(192,1150);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,184,1571,297,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=d,a*b=b*a,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^5>;
 
// generators/relations
 

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