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G = C2×Q83Dic3order 192 = 26·3

Direct product of C2 and Q83Dic3

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Q83Dic3, C63C4≀C2, (C6×D4)⋊8C4, (C6×Q8)⋊8C4, C4○D45Dic3, C4○D4.52D6, (C2×D4)⋊8Dic3, (C2×Q8)⋊8Dic3, Q86(C2×Dic3), D45(C2×Dic3), C12.452(C2×D4), (C2×C12).197D4, C12.85(C22×C4), (C22×C4).373D6, (C22×C6).113D4, C12.99(C22⋊C4), (C2×C12).481C23, (C4×Dic3)⋊63C22, C23.73(C3⋊D4), C4.Dic323C22, C4.15(C22×Dic3), C4.33(C6.D4), (C22×C12).207C22, C22.5(C6.D4), C34(C2×C4≀C2), (C3×C4○D4)⋊3C4, (C2×C4×Dic3)⋊4C2, (C3×D4)⋊18(C2×C4), (C3×Q8)⋊17(C2×C4), (C6×C4○D4).4C2, (C2×C6).39(C2×D4), (C2×C4○D4).10S3, C4.143(C2×C3⋊D4), C6.84(C2×C22⋊C4), (C2×C12).125(C2×C4), (C2×C4.Dic3)⋊21C2, (C2×C4).54(C2×Dic3), C22.11(C2×C3⋊D4), (C2×C4).282(C3⋊D4), (C2×C4).566(C22×S3), C2.20(C2×C6.D4), (C3×C4○D4).41C22, (C2×C6).117(C22⋊C4), SmallGroup(192,794)

Series: Derived Chief Lower central Upper central

C1C12 — C2×Q83Dic3
C1C3C6C12C2×C12C4.Dic3C2×C4.Dic3 — C2×Q83Dic3
C3C6C12 — C2×Q83Dic3
C1C2×C4C22×C4C2×C4○D4

Generators and relations for C2×Q83Dic3
 G = < a,b,c,d,e | a2=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >

Subgroups: 360 in 170 conjugacy classes, 71 normal (43 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C4 [×6], C22 [×3], C22 [×6], C6, C6 [×2], C6 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, Dic3 [×4], C12 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×6], C42 [×3], C2×C8, M4(2) [×3], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C3⋊C8 [×2], C2×Dic3 [×6], C2×C12 [×6], C2×C12 [×5], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×C6, C22×C6, C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3 [×2], C4.Dic3, C4×Dic3 [×2], C4×Dic3, C22×Dic3, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4 [×4], C3×C4○D4 [×2], C2×C4≀C2, Q83Dic3 [×4], C2×C4.Dic3, C2×C4×Dic3, C6×C4○D4, C2×Q83Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C4≀C2 [×2], C2×C22⋊C4, C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C2×C4≀C2, Q83Dic3 [×2], C2×C6.D4, C2×Q83Dic3

Smallest permutation representation of C2×Q83Dic3
On 48 points
Generators in S48
(1 11)(2 12)(3 10)(4 22)(5 23)(6 24)(7 19)(8 20)(9 21)(13 18)(14 16)(15 17)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)(37 44)(38 45)(39 46)(40 47)(41 48)(42 43)
(1 22 13 19)(2 23 14 20)(3 24 15 21)(4 18 7 11)(5 16 8 12)(6 17 9 10)(25 46 28 43)(26 47 29 44)(27 48 30 45)(31 40 34 37)(32 41 35 38)(33 42 36 39)
(1 42 13 39)(2 40 14 37)(3 38 15 41)(4 28 7 25)(5 26 8 29)(6 30 9 27)(10 45 17 48)(11 43 18 46)(12 47 16 44)(19 36 22 33)(20 34 23 31)(21 32 24 35)
(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)
(2 3)(5 6)(8 9)(10 12)(14 15)(16 17)(20 21)(23 24)(25 43 28 46)(26 48 29 45)(27 47 30 44)(31 41 34 38)(32 40 35 37)(33 39 36 42)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G 3 4A4B4C4D4E4F4G4H4I···4P6A6B6C6D···6I8A8B8C8D12A12B12C12D12E···12J
order122222223444444444···46666···688881212121212···12
size111122442111122446···62224···41212121222224···4

48 irreducible representations

dim11111111222222222224
type+++++++++---+
imageC1C2C2C2C2C4C4C4S3D4D4D6Dic3Dic3Dic3D6C3⋊D4C3⋊D4C4≀C2Q83Dic3
kernelC2×Q83Dic3Q83Dic3C2×C4.Dic3C2×C4×Dic3C6×C4○D4C6×D4C6×Q8C3×C4○D4C2×C4○D4C2×C12C22×C6C22×C4C2×D4C2×Q8C4○D4C4○D4C2×C4C23C6C2
# reps14111224131111226284

Matrix representation of C2×Q83Dic3 in GL5(𝔽73)

720000
01000
00100
00010
00001
,
10000
0462700
002700
000720
000072
,
10000
027000
0544600
0004313
0006030
,
720000
072100
00100
0007272
00010
,
270000
0276000
00100
000720
00011

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,46,0,0,0,0,27,27,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,27,54,0,0,0,0,46,0,0,0,0,0,43,60,0,0,0,13,30],[72,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,72,1,0,0,0,72,0],[27,0,0,0,0,0,27,0,0,0,0,60,1,0,0,0,0,0,72,1,0,0,0,0,1] >;

C2×Q83Dic3 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes_3{\rm Dic}_3
% in TeX

G:=Group("C2xQ8:3Dic3");
// GroupNames label

G:=SmallGroup(192,794);
// by ID

G=gap.SmallGroup(192,794);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,136,1684,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^6=1,c^2=b^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations

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