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G = C2xC4xC4oD4order 128 = 27

Direct product of C2xC4 and C4oD4

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

Aliases: C2xC4xC4oD4, C22.10C25, C24.603C23, C42.749C23, C23.104C24, D4o(C2xC42), Q8o(C2xC42), C42o2(C4xD4), C42o2(C2xD4), C42o2(C2xQ8), C42o2(C4xQ8), C42o(C4oD4), D4:7(C22xC4), C2.6(C24xC4), Q8:7(C22xC4), C42o(C22xD4), C42o(C22xQ8), C4.36(C23xC4), (C4xD4):111C22, C4:C4.512C23, (C2xC4).693C24, (C22xC42):20C2, (C2xC42):88C22, (C4xQ8):101C22, C22.1(C23xC4), (C2xD4).495C23, (C2xQ8).478C23, C42o2(C42:C2), C22:C4.124C23, (C23xC4).661C22, C23.154(C22xC4), C42:C2:103C22, (C22xC4).1292C23, (C22xD4).612C22, (C22xQ8).512C22, C4o2(C2xC4xD4), C4o2(C2xC4xQ8), C4o(C4xC4oD4), C42o(C2xC4xD4), (C2xC4)o3(C4xD4), C42o(C2xC4xQ8), (C2xC4xD4):96C2, (C2xC4)o3(C4xQ8), (C2xC4xQ8):64C2, C4:C4o2(C2xC42), C42o4(C2xC4:C4), C42o(C4xC4oD4), (C2xC42)o(C4xD4), (C2xC42)o(C4xQ8), (C2xQ8)o(C2xC42), (C2xD4):54(C2xC4), (C2xQ8):45(C2xC4), C42o2(C2xC4oD4), (C2xD4)o2(C2xC42), C4o2(C2xC42:C2), (C22xC4):48(C2xC4), (C2xC4):10(C22xC4), C22:C4o2(C2xC42), C42o3(C2xC22:C4), C2.4(C22xC4oD4), C42o(C22xC4oD4), (C2xC42)o(C22xQ8), (C2xC4)o3(C42:C2), (C2xC42:C2):69C2, (C2xC4:C4).981C22, C42o2(C2xC42:C2), (C22xC4oD4).30C2, C22.145(C2xC4oD4), (C2xC42)o(C42:C2), (C2xC4oD4).340C22, (C2xC22:C4).560C22, (C2xC4)o2(C2xC4xQ8), (C2xC4)o(C4xC4oD4), (C2xC42)o(C2xC4xQ8), (C2xC42)o(C2xC4:C4), (C2xC42)o(C2xC4oD4), (C2xC4)o2(C2xC42:C2), (C2xC42)o(C22xC4oD4), (C2xC42)o(C2xC42:C2), SmallGroup(128,2156)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C2xC4xC4oD4
C1C2C22C23C22xC4C2xC42C22xC42 — C2xC4xC4oD4
C1C2 — C2xC4xC4oD4
C1C2xC42 — C2xC4xC4oD4
C1C22 — C2xC4xC4oD4

Generators and relations for C2xC4xC4oD4
 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, C2, C4, C4, C22, C22, C22, C2xC4, C2xC4, D4, Q8, C23, C23, C23, C42, C22:C4, C4:C4, C22xC4, C22xC4, C22xC4, C2xD4, C2xQ8, C4oD4, C24, C2xC42, C2xC42, C2xC22:C4, C2xC4:C4, C42:C2, C4xD4, C4xQ8, C23xC4, C22xD4, C22xQ8, C2xC4oD4, C22xC42, C2xC42:C2, C2xC4xD4, C2xC4xQ8, C4xC4oD4, C22xC4oD4, C2xC4xC4oD4
Quotients: C1, C2, C4, C22, C2xC4, C23, C22xC4, C4oD4, C24, C23xC4, C2xC4oD4, C25, C4xC4oD4, C24xC4, C22xC4oD4, C2xC4xC4oD4

Smallest permutation representation of C2xC4xC4oD4
On 64 points
Generators in S64
(1 25)(2 26)(3 27)(4 28)(5 37)(6 38)(7 39)(8 40)(9 15)(10 16)(11 13)(12 14)(17 31)(18 32)(19 29)(20 30)(21 35)(22 36)(23 33)(24 34)(41 47)(42 48)(43 45)(44 46)(49 62)(50 63)(51 64)(52 61)(53 59)(54 60)(55 57)(56 58)
(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 7)(2 30 12 8)(3 31 9 5)(4 32 10 6)(13 39 25 19)(14 40 26 20)(15 37 27 17)(16 38 28 18)(21 57 52 45)(22 58 49 46)(23 59 50 47)(24 60 51 48)(33 53 63 41)(34 54 64 42)(35 55 61 43)(36 56 62 44)
(1 19 11 39)(2 20 12 40)(3 17 9 37)(4 18 10 38)(5 27 31 15)(6 28 32 16)(7 25 29 13)(8 26 30 14)(21 43 52 55)(22 44 49 56)(23 41 50 53)(24 42 51 54)(33 47 63 59)(34 48 64 60)(35 45 61 57)(36 46 62 58)
(1 35)(2 36)(3 33)(4 34)(5 41)(6 42)(7 43)(8 44)(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 55)(30 56)(31 53)(32 54)(37 47)(38 48)(39 45)(40 46)

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

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

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

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+++++++
imageC1C2C2C2C2C2C2C4C4oD4
kernelC2xC4xC4oD4C22xC42C2xC42:C2C2xC4xD4C2xC4xQ8C4xC4oD4C22xC4oD4C2xC4oD4C2xC4
# reps133621613216

Matrix representation of C2xC4xC4oD4 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] >;

C2xC4xC4oD4 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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