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G = C2×C4×C4○D4order 128 = 27

Direct product of C2×C4 and C4○D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C2×C4×C4○D4, C22.10C25, C24.603C23, C42.749C23, C23.104C24, D4(C2×C42), Q8(C2×C42), C422(C4×D4), C422(C2×D4), C422(C2×Q8), C422(C4×Q8), C42(C4○D4), D47(C22×C4), C2.6(C24×C4), Q87(C22×C4), C42(C22×D4), C42(C22×Q8), C4.36(C23×C4), (C4×D4)⋊111C22, C4⋊C4.512C23, (C2×C4).693C24, (C22×C42)⋊20C2, (C2×C42)⋊88C22, (C4×Q8)⋊101C22, C22.1(C23×C4), (C2×D4).495C23, (C2×Q8).478C23, C422(C42⋊C2), C22⋊C4.124C23, (C23×C4).661C22, C23.154(C22×C4), C42⋊C2103C22, (C22×C4).1292C23, (C22×D4).612C22, (C22×Q8).512C22, C42(C2×C4×D4), C42(C2×C4×Q8), C4(C4×C4○D4), C42(C2×C4×D4), (C2×C4)3(C4×D4), C42(C2×C4×Q8), (C2×C4×D4)⋊96C2, (C2×C4)3(C4×Q8), (C2×C4×Q8)⋊64C2, C4⋊C42(C2×C42), C424(C2×C4⋊C4), C42(C4×C4○D4), (C2×C42)(C4×D4), (C2×C42)(C4×Q8), (C2×Q8)(C2×C42), (C2×D4)⋊54(C2×C4), (C2×Q8)⋊45(C2×C4), C422(C2×C4○D4), (C2×D4)2(C2×C42), C42(C2×C42⋊C2), (C22×C4)⋊48(C2×C4), (C2×C4)⋊10(C22×C4), C22⋊C42(C2×C42), C423(C2×C22⋊C4), C2.4(C22×C4○D4), C42(C22×C4○D4), (C2×C42)(C22×Q8), (C2×C4)3(C42⋊C2), (C2×C42⋊C2)⋊69C2, (C2×C4⋊C4).981C22, C422(C2×C42⋊C2), (C22×C4○D4).30C2, C22.145(C2×C4○D4), (C2×C42)(C42⋊C2), (C2×C4○D4).340C22, (C2×C22⋊C4).560C22, (C2×C4)2(C2×C4×Q8), (C2×C4)(C4×C4○D4), (C2×C42)(C2×C4×Q8), (C2×C42)(C2×C4⋊C4), (C2×C42)(C2×C4○D4), (C2×C4)2(C2×C42⋊C2), (C2×C42)(C22×C4○D4), (C2×C42)(C2×C42⋊C2), SmallGroup(128,2156)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2×C4×C4○D4
C1C2C22C23C22×C4C2×C42C22×C42 — C2×C4×C4○D4
C1C2 — C2×C4×C4○D4
C1C2×C42 — C2×C4×C4○D4
C1C22 — C2×C4×C4○D4

Generators and relations for C2×C4×C4○D4
 G = < a,b,c,d,e | a2=b4=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 1020 in 864 conjugacy classes, 708 normal (9 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×24], C4 [×12], C22, C22 [×18], C22 [×36], C2×C4 [×96], C2×C4 [×60], D4 [×48], Q8 [×16], C23, C23 [×18], C23 [×12], C42 [×40], C22⋊C4 [×24], C4⋊C4 [×24], C22×C4, C22×C4 [×71], C22×C4 [×24], C2×D4 [×36], C2×Q8 [×12], C4○D4 [×64], C24 [×3], C2×C42, C2×C42 [×33], C2×C22⋊C4 [×6], C2×C4⋊C4 [×6], C42⋊C2 [×24], C4×D4 [×48], C4×Q8 [×16], C23×C4 [×9], C22×D4 [×3], C22×Q8, C2×C4○D4 [×24], C22×C42 [×3], C2×C42⋊C2 [×3], C2×C4×D4 [×6], C2×C4×Q8 [×2], C4×C4○D4 [×16], C22×C4○D4, C2×C4×C4○D4
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C4○D4 [×8], C24 [×31], C23×C4 [×30], C2×C4○D4 [×12], C25, C4×C4○D4 [×4], C24×C4, C22×C4○D4 [×2], C2×C4×C4○D4

Smallest permutation representation of C2×C4×C4○D4
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 39)(6 40)(7 37)(8 38)(9 13)(10 14)(11 15)(12 16)(17 31)(18 32)(19 29)(20 30)(21 33)(22 34)(23 35)(24 36)(41 45)(42 46)(43 47)(44 48)(49 64)(50 61)(51 62)(52 63)(53 57)(54 58)(55 59)(56 60)
(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)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 29 11 5)(2 30 12 6)(3 31 9 7)(4 32 10 8)(13 37 27 17)(14 38 28 18)(15 39 25 19)(16 40 26 20)(21 57 50 47)(22 58 51 48)(23 59 52 45)(24 60 49 46)(33 53 61 43)(34 54 62 44)(35 55 63 41)(36 56 64 42)
(1 19 11 39)(2 20 12 40)(3 17 9 37)(4 18 10 38)(5 25 29 15)(6 26 30 16)(7 27 31 13)(8 28 32 14)(21 43 50 53)(22 44 51 54)(23 41 52 55)(24 42 49 56)(33 47 61 57)(34 48 62 58)(35 45 63 59)(36 46 64 60)
(1 33)(2 34)(3 35)(4 36)(5 43)(6 44)(7 41)(8 42)(9 63)(10 64)(11 61)(12 62)(13 52)(14 49)(15 50)(16 51)(17 59)(18 60)(19 57)(20 58)(21 25)(22 26)(23 27)(24 28)(29 53)(30 54)(31 55)(32 56)(37 45)(38 46)(39 47)(40 48)

G:=sub<Sym(64)| (1,25)(2,26)(3,27)(4,28)(5,39)(6,40)(7,37)(8,38)(9,13)(10,14)(11,15)(12,16)(17,31)(18,32)(19,29)(20,30)(21,33)(22,34)(23,35)(24,36)(41,45)(42,46)(43,47)(44,48)(49,64)(50,61)(51,62)(52,63)(53,57)(54,58)(55,59)(56,60), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,29,11,5)(2,30,12,6)(3,31,9,7)(4,32,10,8)(13,37,27,17)(14,38,28,18)(15,39,25,19)(16,40,26,20)(21,57,50,47)(22,58,51,48)(23,59,52,45)(24,60,49,46)(33,53,61,43)(34,54,62,44)(35,55,63,41)(36,56,64,42), (1,19,11,39)(2,20,12,40)(3,17,9,37)(4,18,10,38)(5,25,29,15)(6,26,30,16)(7,27,31,13)(8,28,32,14)(21,43,50,53)(22,44,51,54)(23,41,52,55)(24,42,49,56)(33,47,61,57)(34,48,62,58)(35,45,63,59)(36,46,64,60), (1,33)(2,34)(3,35)(4,36)(5,43)(6,44)(7,41)(8,42)(9,63)(10,64)(11,61)(12,62)(13,52)(14,49)(15,50)(16,51)(17,59)(18,60)(19,57)(20,58)(21,25)(22,26)(23,27)(24,28)(29,53)(30,54)(31,55)(32,56)(37,45)(38,46)(39,47)(40,48)>;

G:=Group( (1,25)(2,26)(3,27)(4,28)(5,39)(6,40)(7,37)(8,38)(9,13)(10,14)(11,15)(12,16)(17,31)(18,32)(19,29)(20,30)(21,33)(22,34)(23,35)(24,36)(41,45)(42,46)(43,47)(44,48)(49,64)(50,61)(51,62)(52,63)(53,57)(54,58)(55,59)(56,60), (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)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,29,11,5)(2,30,12,6)(3,31,9,7)(4,32,10,8)(13,37,27,17)(14,38,28,18)(15,39,25,19)(16,40,26,20)(21,57,50,47)(22,58,51,48)(23,59,52,45)(24,60,49,46)(33,53,61,43)(34,54,62,44)(35,55,63,41)(36,56,64,42), (1,19,11,39)(2,20,12,40)(3,17,9,37)(4,18,10,38)(5,25,29,15)(6,26,30,16)(7,27,31,13)(8,28,32,14)(21,43,50,53)(22,44,51,54)(23,41,52,55)(24,42,49,56)(33,47,61,57)(34,48,62,58)(35,45,63,59)(36,46,64,60), (1,33)(2,34)(3,35)(4,36)(5,43)(6,44)(7,41)(8,42)(9,63)(10,64)(11,61)(12,62)(13,52)(14,49)(15,50)(16,51)(17,59)(18,60)(19,57)(20,58)(21,25)(22,26)(23,27)(24,28)(29,53)(30,54)(31,55)(32,56)(37,45)(38,46)(39,47)(40,48) );

G=PermutationGroup([(1,25),(2,26),(3,27),(4,28),(5,39),(6,40),(7,37),(8,38),(9,13),(10,14),(11,15),(12,16),(17,31),(18,32),(19,29),(20,30),(21,33),(22,34),(23,35),(24,36),(41,45),(42,46),(43,47),(44,48),(49,64),(50,61),(51,62),(52,63),(53,57),(54,58),(55,59),(56,60)], [(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),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,29,11,5),(2,30,12,6),(3,31,9,7),(4,32,10,8),(13,37,27,17),(14,38,28,18),(15,39,25,19),(16,40,26,20),(21,57,50,47),(22,58,51,48),(23,59,52,45),(24,60,49,46),(33,53,61,43),(34,54,62,44),(35,55,63,41),(36,56,64,42)], [(1,19,11,39),(2,20,12,40),(3,17,9,37),(4,18,10,38),(5,25,29,15),(6,26,30,16),(7,27,31,13),(8,28,32,14),(21,43,50,53),(22,44,51,54),(23,41,52,55),(24,42,49,56),(33,47,61,57),(34,48,62,58),(35,45,63,59),(36,46,64,60)], [(1,33),(2,34),(3,35),(4,36),(5,43),(6,44),(7,41),(8,42),(9,63),(10,64),(11,61),(12,62),(13,52),(14,49),(15,50),(16,51),(17,59),(18,60),(19,57),(20,58),(21,25),(22,26),(23,27),(24,28),(29,53),(30,54),(31,55),(32,56),(37,45),(38,46),(39,47),(40,48)])

80 conjugacy classes

class 1 2A···2G2H···2S4A···4X4Y···4BH
order12···22···24···44···4
size11···12···21···12···2

80 irreducible representations

dim111111112
type+++++++
imageC1C2C2C2C2C2C2C4C4○D4
kernelC2×C4×C4○D4C22×C42C2×C42⋊C2C2×C4×D4C2×C4×Q8C4×C4○D4C22×C4○D4C2×C4○D4C2×C4
# reps133621613216

Matrix representation of C2×C4×C4○D4 in GL4(𝔽5) generated by

4000
0400
0040
0004
,
3000
0100
0020
0002
,
4000
0100
0020
0002
,
4000
0100
0030
0002
,
1000
0400
0002
0030
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,1,0,0,0,0,3,0,0,0,0,2],[1,0,0,0,0,4,0,0,0,0,0,3,0,0,2,0] >;

C2×C4×C4○D4 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_4\circ D_4
% in TeX

G:=Group("C2xC4xC4oD4");
// GroupNames label

G:=SmallGroup(128,2156);
// by ID

G=gap.SmallGroup(128,2156);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,352,136]);
// Polycyclic

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

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