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G = C23.26D6order 96 = 25·3

2nd non-split extension by C23 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.26D6, (C2×C12)⋊6C4, (C2×C4)⋊4Dic3, C12.37(C2×C4), C43(C4⋊Dic3), C4⋊Dic317C2, (C2×C4).102D6, (C22×C4).9S3, C34(C42⋊C2), (C4×Dic3)⋊15C2, C4(C6.D4), C2.4(C4○D12), C6.16(C4○D4), (C2×C6).44C23, C6.24(C22×C4), C4.15(C2×Dic3), (C2×C12).93C22, (C22×C12).10C2, C6.D4.5C2, C22.5(C2×Dic3), C2.5(C22×Dic3), (C22×C6).36C22, C22.22(C22×S3), (C2×Dic3).35C22, (C2×C6).35(C2×C4), (C2×C4)(C4⋊Dic3), SmallGroup(96,133)

Series: Derived Chief Lower central Upper central

C1C6 — C23.26D6
C1C3C6C2×C6C2×Dic3C4×Dic3 — C23.26D6
C3C6 — C23.26D6
C1C2×C4C22×C4

Generators and relations for C23.26D6
 G = < a,b,c,d,e | a2=b2=c2=1, d6=c, e2=cb=bc, ab=ba, eae-1=ac=ca, ad=da, bd=db, be=eb, cd=dc, ce=ec, ede-1=d5 >

Subgroups: 130 in 76 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, Dic3 [×4], C12 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C22×C6, C42⋊C2, C4×Dic3 [×2], C4⋊Dic3 [×2], C6.D4 [×2], C22×C12, C23.26D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], C22×C4, C4○D4 [×2], C2×Dic3 [×6], C22×S3, C42⋊C2, C4○D12 [×2], C22×Dic3, C23.26D6

Smallest permutation representation of C23.26D6
On 48 points
Generators in S48
(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 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(25 44)(26 45)(27 46)(28 47)(29 48)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(36 43)
(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 37 17 36)(2 42 18 29)(3 47 19 34)(4 40 20 27)(5 45 21 32)(6 38 22 25)(7 43 23 30)(8 48 24 35)(9 41 13 28)(10 46 14 33)(11 39 15 26)(12 44 16 31)

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

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

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

C23.26D6 is a maximal subgroup of
C12.8C42  C423Dic3  C12.3C42  (C2×C24)⋊C4  C12.20C42  M4(2)⋊4Dic3  C24⋊C4⋊C2  C23.15D12  C23.18D12  C4⋊C4.225D6  C4⋊C4.234D6  (C2×D12)⋊13C4  C6.Q16⋊C2  (C2×Q8).51D6  C12.12C42  Dic3⋊C8⋊C2  C23.27D12  C23.28D12  C12.88(C2×Q8)  C23.51D12  C23.52D12  C12.7C42  C23.53D12  M4(2)⋊24D6  (C6×D4)⋊6C4  (C6×Q8)⋊6C4  C4○D44Dic3  (C6×D4)⋊9C4  (C6×D4)⋊10C4  C42.274D6  C4×C4○D12  C24.42D6  C6.72+ 1+4  C6.82+ 1+4  C6.52- 1+4  C42.87D6  C42.88D6  C42.90D6  S3×C42⋊C2  C429D6  C42.102D6  C42.105D6  C42.106D6  C42.229D6  C42.117D6  C42.119D6  C6.712- 1+4  C6.432+ 1+4  C6.452+ 1+4  C6.472+ 1+4  C6.152- 1+4  C6.212- 1+4  C6.232- 1+4  C6.242- 1+4  C6.802- 1+4  C6.1222+ 1+4  C24.83D6  C24.49D6  C24.52D6  C6.422- 1+4  C6.442- 1+4  C6.1052- 1+4  Dic3×C4○D4  C6.1442+ 1+4  (C2×D4)⋊43D6  C6.1462+ 1+4  C23.26D18  C62.11C23  C62.25C23  C62.97C23  C62.247C23  (D5×C12)⋊C4  (C4×D5)⋊Dic3  (C6×Dic5)⋊7C4  C23.26D30  (C2×C12)⋊6F5
C23.26D6 is a maximal quotient of
C42.285D6  C42.270D6  C426Dic3  C4×C4⋊Dic3  C4211Dic3  C427Dic3  C24.19D6  C4⋊C45Dic3  C42.187D6  C4×C6.D4  C24.74D6  C24.75D6  C23.26D18  C62.11C23  C62.25C23  C62.97C23  C62.247C23  (D5×C12)⋊C4  (C4×D5)⋊Dic3  (C6×Dic5)⋊7C4  C23.26D30  (C2×C12)⋊6F5

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F4G···4N6A···6G12A···12H
order12222234444444···46···612···12
size11112221111226···62···22···2

36 irreducible representations

dim111111222222
type++++++-++
imageC1C2C2C2C2C4S3Dic3D6D6C4○D4C4○D12
kernelC23.26D6C4×Dic3C4⋊Dic3C6.D4C22×C12C2×C12C22×C4C2×C4C2×C4C23C6C2
# reps122218142148

Matrix representation of C23.26D6 in GL3(𝔽13) generated by

1200
010
0012
,
1200
0120
0012
,
100
0120
0012
,
100
070
0011
,
800
0011
060
G:=sub<GL(3,GF(13))| [12,0,0,0,1,0,0,0,12],[12,0,0,0,12,0,0,0,12],[1,0,0,0,12,0,0,0,12],[1,0,0,0,7,0,0,0,11],[8,0,0,0,0,6,0,11,0] >;

C23.26D6 in GAP, Magma, Sage, TeX

C_2^3._{26}D_6
% in TeX

G:=Group("C2^3.26D6");
// GroupNames label

G:=SmallGroup(96,133);
// by ID

G=gap.SmallGroup(96,133);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,103,362,2309]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=c,e^2=c*b=b*c,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^5>;
// generators/relations

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