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G = C23.8Dic6order 192 = 26·3

6th non-split extension by C23 of Dic6 acting via Dic6/C6=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.8Dic6, M4(2).30D6, C12.37(C4⋊C4), (C2×C12).28Q8, C12.441(C2×D4), (C2×C12).483D4, (C2×C4).16Dic6, C4.Dic3.6C4, (C22×C6).16Q8, C12.68(C22×C4), (C22×C4).150D6, C12.53D413C2, C22.4(C2×Dic6), C32(M4(2).C4), C4.21(Dic3⋊C4), (C2×C12).415C23, (C6×M4(2)).27C2, (C2×M4(2)).16S3, C4.Dic3.41C22, C22.17(Dic3⋊C4), (C22×C12).183C22, (C3×M4(2)).33C22, C3⋊C8.5(C2×C4), C4.90(S3×C2×C4), C6.52(C2×C4⋊C4), (C2×C4).49(C4×S3), (C2×C6).11(C2×Q8), (C2×C6).54(C4⋊C4), C4.131(C2×C3⋊D4), (C2×C12).103(C2×C4), (C2×C3⋊C8).143C22, C2.19(C2×Dic3⋊C4), (C2×C4).194(C3⋊D4), (C2×C4).511(C22×S3), (C2×C4.Dic3).24C2, SmallGroup(192,683)

Series: Derived Chief Lower central Upper central

C1C12 — C23.8Dic6
C1C3C6C12C2×C12C2×C3⋊C8C2×C4.Dic3 — C23.8Dic6
C3C6C12 — C23.8Dic6
C1C4C22×C4C2×M4(2)

Generators and relations for C23.8Dic6
 G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=bd6, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd11 >

Subgroups: 184 in 102 conjugacy classes, 59 normal (39 characteristic)
C1, C2, C2 [×3], C3, C4 [×4], C22 [×3], C22, C6, C6 [×3], C8 [×8], C2×C4 [×6], C23, C12 [×4], C2×C6 [×3], C2×C6, C2×C8 [×4], M4(2) [×2], M4(2) [×8], C22×C4, C3⋊C8 [×4], C3⋊C8 [×2], C24 [×2], C2×C12 [×6], C22×C6, C8.C4 [×4], C2×M4(2), C2×M4(2) [×2], C2×C3⋊C8 [×2], C2×C3⋊C8, C4.Dic3 [×6], C4.Dic3, C2×C24, C3×M4(2) [×2], C3×M4(2), C22×C12, M4(2).C4, C12.53D4 [×4], C2×C4.Dic3 [×2], C6×M4(2), C23.8Dic6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, C2×C3⋊D4, M4(2).C4, C2×Dic3⋊C4, C23.8Dic6

Smallest permutation representation of C23.8Dic6
On 48 points
Generators in S48
(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)
(2 14)(4 16)(6 18)(8 20)(10 22)(12 24)(26 38)(28 40)(30 42)(32 44)(34 46)(36 48)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 31 7 37 13 43 19 25)(2 42 20 36 14 30 8 48)(3 41 9 47 15 29 21 35)(4 28 22 46 16 40 10 34)(5 27 11 33 17 39 23 45)(6 38 24 32 18 26 12 44)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B6C6D6E8A8B8C8D8E···8L12A12B12C12D12E12F24A···24H
order122223444446666688888···812121212121224···24
size1122221122222244444412···122222444···4

42 irreducible representations

dim11111222222222244
type++++++--++--
imageC1C2C2C2C4S3D4Q8Q8D6D6Dic6C4×S3C3⋊D4Dic6M4(2).C4C23.8Dic6
kernelC23.8Dic6C12.53D4C2×C4.Dic3C6×M4(2)C4.Dic3C2×M4(2)C2×C12C2×C12C22×C6M4(2)C22×C4C2×C4C2×C4C2×C4C23C3C1
# reps14218121121244224

Matrix representation of C23.8Dic6 in GL6(𝔽73)

100000
010000
001000
000100
0010720
00720072
,
7200000
0720000
001000
00727200
000010
00720072
,
100000
010000
0072000
0007200
0000720
0000072
,
6400000
13650000
00727100
0060100
005972046
0011720
,
33250000
44400000
0010710
000011
00140720
00132710

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,72,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,72,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[64,13,0,0,0,0,0,65,0,0,0,0,0,0,72,60,59,1,0,0,71,1,72,1,0,0,0,0,0,72,0,0,0,0,46,0],[33,44,0,0,0,0,25,40,0,0,0,0,0,0,1,0,14,13,0,0,0,0,0,27,0,0,71,1,72,1,0,0,0,1,0,0] >;

C23.8Dic6 in GAP, Magma, Sage, TeX

C_2^3._8{\rm Dic}_6
% in TeX

G:=Group("C2^3.8Dic6");
// GroupNames label

G:=SmallGroup(192,683);
// by ID

G=gap.SmallGroup(192,683);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,477,422,58,136,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=b*d^6,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^11>;
// generators/relations

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