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G = D247C4order 192 = 26·3

7th semidirect product of D24 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D247C4, Dic127C4, M4(2).27D6, C33(C8○D8), C3⋊C8.38D4, C24⋊C26C4, C8.16(C4×S3), C6.56(C4×D4), C8.C48S3, C24.35(C2×C4), (C8×Dic3)⋊1C2, C4○D24.5C2, C4.213(S3×D4), (C2×C8).252D6, D12.C411C2, D12.11(C2×C4), C12.372(C2×D4), D12⋊C412C2, C12.55(C22×C4), (C2×C24).42C22, Dic6.11(C2×C4), (C2×C12).311C23, C4○D12.18C22, C2.16(Dic35D4), C22.2(Q83S3), (C4×Dic3).235C22, (C3×M4(2)).29C22, C4.47(S3×C2×C4), (C3×C8.C4)⋊5C2, (C2×C6).2(C4○D4), (C2×C3⋊C8).238C22, (C2×C4).414(C22×S3), SmallGroup(192,454)

Series: Derived Chief Lower central Upper central

C1C12 — D247C4
C1C3C6C12C2×C12C4○D12C4○D24 — D247C4
C3C6C12 — D247C4
C1C4C2×C4C8.C4

Generators and relations for D247C4
 G = < a,b,c | a24=b2=c4=1, bab=a-1, cac-1=a17, cbc-1=a22b >

Subgroups: 272 in 106 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C8 [×4], C2×C4, C2×C4 [×3], D4 [×4], Q8 [×2], Dic3 [×4], C12 [×2], D6 [×2], C2×C6, C42, C2×C8, C2×C8 [×3], M4(2) [×2], M4(2) [×2], D8, SD16 [×2], Q16, C4○D4 [×2], C3⋊C8 [×2], C24 [×2], C24 [×2], Dic6 [×2], C4×S3 [×2], D12 [×2], C2×Dic3, C3⋊D4 [×2], C2×C12, C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, S3×C8 [×2], C8⋊S3 [×2], C24⋊C2 [×2], D24, Dic12, C2×C3⋊C8, C4×Dic3, C2×C24, C3×M4(2) [×2], C4○D12 [×2], C8○D8, C8×Dic3, D12⋊C4 [×2], C3×C8.C4, C4○D24, D12.C4 [×2], D247C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], C23, D6 [×3], C22×C4, C2×D4, C4○D4, C4×S3 [×2], C22×S3, C4×D4, S3×C2×C4, S3×D4, Q83S3, C8○D8, Dic35D4, D247C4

Smallest permutation representation of D247C4
On 48 points
Generators in S48
(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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 48)
(1 19 13 7)(2 12 14 24)(3 5 15 17)(4 22 16 10)(6 8 18 20)(9 11 21 23)(25 29)(26 46)(27 39)(28 32)(30 42)(31 35)(33 45)(34 38)(36 48)(37 41)(40 44)(43 47)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E4F4G4H4I6A6B8A8B8C8D8E8F8G8H8I8J8K8L8M8N12A12B12C24A24B24C24D24E24F24G24H
order12222344444444466888888888888881212122424242424242424
size1121212211266661212242222333344446622444448888

42 irreducible representations

dim1111111112222222444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4S3D4D6D6C4○D4C4×S3C8○D8S3×D4Q83S3D247C4
kernelD247C4C8×Dic3D12⋊C4C3×C8.C4C4○D24D12.C4C24⋊C2D24Dic12C8.C4C3⋊C8C2×C8M4(2)C2×C6C8C3C4C22C1
# reps1121124221212248114

Matrix representation of D247C4 in GL4(𝔽73) generated by

10000
02200
0001
007272
,
07200
72000
0001
0010
,
27000
07200
0010
007272
G:=sub<GL(4,GF(73))| [10,0,0,0,0,22,0,0,0,0,0,72,0,0,1,72],[0,72,0,0,72,0,0,0,0,0,0,1,0,0,1,0],[27,0,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;

D247C4 in GAP, Magma, Sage, TeX

D_{24}\rtimes_7C_4
% in TeX

G:=Group("D24:7C4");
// GroupNames label

G:=SmallGroup(192,454);
// by ID

G=gap.SmallGroup(192,454);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,555,58,136,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^22*b>;
// generators/relations

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