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G = C22×C3⋊D4order 96 = 25·3

Direct product of C22 and C3⋊D4

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

Aliases: C22×C3⋊D4, C245S3, C235D6, D63C23, C6.15C24, Dic32C23, (C2×C6)⋊9D4, C63(C2×D4), C33(C22×D4), (C2×C6)⋊3C23, (C23×C6)⋊4C2, (S3×C23)⋊5C2, C2.15(S3×C23), (C22×C6)⋊7C22, C223(C22×S3), (C22×S3)⋊7C22, (C22×Dic3)⋊9C2, (C2×Dic3)⋊11C22, SmallGroup(96,219)

Series: Derived Chief Lower central Upper central

C1C6 — C22×C3⋊D4
C1C3C6D6C22×S3S3×C23 — C22×C3⋊D4
C3C6 — C22×C3⋊D4
C1C23C24

Generators and relations for C22×C3⋊D4
 G = < a,b,c,d,e | a2=b2=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 498 in 236 conjugacy classes, 105 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C22 [×11], C22 [×28], S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×6], D4 [×16], C23, C23 [×6], C23 [×14], Dic3 [×4], D6 [×4], D6 [×12], C2×C6 [×11], C2×C6 [×12], C22×C4, C2×D4 [×12], C24, C24, C2×Dic3 [×6], C3⋊D4 [×16], C22×S3 [×6], C22×S3 [×4], C22×C6, C22×C6 [×6], C22×C6 [×4], C22×D4, C22×Dic3, C2×C3⋊D4 [×12], S3×C23, C23×C6, C22×C3⋊D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C3⋊D4 [×6], S3×C23, C22×C3⋊D4

Smallest permutation representation of C22×C3⋊D4
On 48 points
Generators in S48
(1 44)(2 41)(3 42)(4 43)(5 46)(6 47)(7 48)(8 45)(9 18)(10 19)(11 20)(12 17)(13 25)(14 26)(15 27)(16 28)(21 35)(22 36)(23 33)(24 34)(29 40)(30 37)(31 38)(32 39)
(1 5)(2 6)(3 7)(4 8)(9 23)(10 24)(11 21)(12 22)(13 32)(14 29)(15 30)(16 31)(17 36)(18 33)(19 34)(20 35)(25 39)(26 40)(27 37)(28 38)(41 47)(42 48)(43 45)(44 46)
(1 15 20)(2 17 16)(3 13 18)(4 19 14)(5 30 35)(6 36 31)(7 32 33)(8 34 29)(9 42 25)(10 26 43)(11 44 27)(12 28 41)(21 46 37)(22 38 47)(23 48 39)(24 40 45)
(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 46)(2 45)(3 48)(4 47)(5 44)(6 43)(7 42)(8 41)(9 32)(10 31)(11 30)(12 29)(13 23)(14 22)(15 21)(16 24)(17 40)(18 39)(19 38)(20 37)(25 33)(26 36)(27 35)(28 34)

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

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

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

C22×C3⋊D4 is a maximal subgroup of
C24.23D6  C24.24D6  C24.60D6  C24.25D6  C233D12  C24.27D6  C24.76D6  C24.32D6  C24.35D6  C24.38D6  C234D12  C247D6  C248D6  C24.44D6  C24.45D6  C2412D6  C22×S3×D4
C22×C3⋊D4 is a maximal quotient of
C24.83D6  C2412D6  C24.52D6  C24.53D6  C6.442- 1+4  C6.452- 1+4  C12.C24  C6.1042- 1+4  C6.1052- 1+4  (C2×D4)⋊43D6  C6.1452+ 1+4  C6.1462+ 1+4  C6.1072- 1+4  (C2×C12)⋊17D4  C6.1082- 1+4  C6.1482+ 1+4  D12.32C23  D12.33C23  D12.34C23  D12.35C23

36 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O 3 4A4B4C4D6A···6O
order12···222222222344446···6
size11···122226666266662···2

36 irreducible representations

dim111112222
type++++++++
imageC1C2C2C2C2S3D4D6C3⋊D4
kernelC22×C3⋊D4C22×Dic3C2×C3⋊D4S3×C23C23×C6C24C2×C6C23C22
# reps1112111478

Matrix representation of C22×C3⋊D4 in GL6(𝔽13)

100000
010000
0012000
0001200
000010
000001
,
1200000
0120000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000012
0000112
,
010000
1200000
0001200
001000
0000411
000029
,
1200000
010000
0012000
000100
000001
000010

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,4,2,0,0,0,0,11,9],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C22×C3⋊D4 in GAP, Magma, Sage, TeX

C_2^2\times C_3\rtimes D_4
% in TeX

G:=Group("C2^2xC3:D4");
// GroupNames label

G:=SmallGroup(96,219);
// by ID

G=gap.SmallGroup(96,219);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,579,2309]);
// Polycyclic

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

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