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G = C24.46D6  order 192 = 26·3

35th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.46D6, C6.292+ (1+4), C22≀C2⋊7S3, (C2×D4).86D6, C22⋊C4.2D6, D6⋊3D4⋊14C2, C24⋊4S3⋊8C2, D6⋊C4⋊14C22, (D4×Dic3)⋊13C2, C23.14D6⋊5C2, Dic3⋊4D4⋊4C2, C23.9D6⋊14C2, (C2×C12).31C23, (C2×C6).137C24, C4⋊Dic3⋊27C22, C23.12D6⋊12C2, C2.31(D4⋊6D6), Dic3⋊C4⋊12C22, C3⋊3(C22.32C24), (C4×Dic3)⋊17C22, (C2×Dic6)⋊22C22, (C6×D4).111C22, C23.23D6⋊5C2, C23.8D6⋊12C2, (C23×C6).70C22, Dic3.D4⋊14C2, C23.11D6⋊14C2, C6.D4⋊17C22, (C22×S3).56C23, (C22×C6).182C23, C23.187(C22×S3), C22.158(S3×C23), (C2×Dic3).62C23, C22.10(D4⋊2S3), (C22×Dic3)⋊16C22, (S3×C2×C4)⋊10C22, C6.78(C2×C4○D4), (C3×C22≀C2)⋊8C2, (C2×C6).44(C4○D4), C2.29(C2×D4⋊2S3), (C2×C3⋊D4)⋊10C22, (C2×C4).31(C22×S3), (C2×C6.D4)⋊21C2, (C3×C22⋊C4).3C22, SmallGroup(192,1152)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C24.46D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C23.9D6 — C24.46D6
Lower central
C3 — C2×C6 — C24.46D6

Subgroups: 656 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×10], C22, C22 [×2], C22 [×18], S3, C6 [×3], C6 [×5], C2×C4 [×3], C2×C4 [×11], D4 [×9], Q8, C23 [×4], C23 [×5], Dic3 [×7], C12 [×3], D6 [×3], C2×C6, C2×C6 [×2], C2×C6 [×15], C42 [×2], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×3], C2×D4 [×4], C2×Q8, C24, Dic6, C4×S3, C2×Dic3 [×7], C2×Dic3 [×3], C3⋊D4 [×5], C2×C12 [×3], C3×D4 [×4], C22×S3, C22×C6 [×4], C22×C6 [×4], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C22≀C2, C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C42⋊2C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×2], C6.D4 [×9], C3×C22⋊C4 [×3], C2×Dic6, S3×C2×C4, C22×Dic3 [×3], C2×C3⋊D4 [×4], C6×D4 [×3], C23×C6, C22.32C24, Dic3.D4, C23.8D6 [×2], Dic3⋊4D4, C23.9D6, C23.11D6, D4×Dic3, C23.23D6, C23.12D6, D6⋊3D4, C23.14D6 [×2], C2×C6.D4, C24⋊4S3, C3×C22≀C2, C24.46D6

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], D4⋊2S3 [×2], S3×C23, C22.32C24, C2×D4⋊2S3, D4⋊6D6 [×2], C24.46D6

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

Smallest permutation representation
►On 48 points
Generators in S48
(2 20)(4 22)(6 24)(8 14)(10 16)(12 18)(25 41)(26 32)(27 43)(28 34)(29 45)(30 36)(31 47)(33 37)(35 39)(38 44)(40 46)(42 48)

(1 7)(3 9)(5 11)(13 19)(15 21)(17 23)(25 47)(26 42)(27 37)(28 44)(29 39)(30 46)(31 41)(32 48)(33 43)(34 38)(35 45)(36 40)

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

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

Matrix representation ►G ⊆ GL8(𝔽13)

120000000
01000000
00100000
00010000
00001000
000001200
00000010
000000012
,
120000000
01000000
001200000
000120000
00001000
00000100
000000120
000000012
,
120000000
012000000
00100000
00010000
000012000
000001200
000000120
000000012
,
120000000
012000000
00100000
00010000
00001000
00000100
00000010
00000001
,
01000000
120000000
00010000
001210000
00000100
00001000
00000001
00000010
,
80000000
08000000
00330000
006100000
00000001
00000010
00000100
00001000

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H···4L6A6B6C6D···6I6J12A12B12C
order1222222222344444444···46666···66121212
size111122444122444666612···122224···48888

36 irreducible representations

dim1111111111111122222444
type+++++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2S3D6D6D6C4○D42+ (1+4)D4⋊2S3D4⋊6D6
kernelC24.46D6Dic3.D4C23.8D6Dic3⋊4D4C23.9D6C23.11D6D4×Dic3C23.23D6C23.12D6D6⋊3D4C23.14D6C2×C6.D4C24⋊4S3C3×C22≀C2C22≀C2C22⋊C4C2×D4C24C2×C6C6C22C2
# reps1121111111211113314224

In GAP, Magma, Sage, TeX 

C_2^4._{46}D_6
 
% in TeX
 
G:=Group("C2^4.46D6");
 
// GroupNames label
 
G:=SmallGroup(192,1152);
 
// by ID
 
G=gap.SmallGroup(192,1152);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,675,570,6278]);
 
// 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=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,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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