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G = C4×C22⋊C8  order 128 = 27

Direct product of C4 and C22⋊C8

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

Aliases: C4×C22⋊C8, C42.457D4, C23.33C42, (C22×C4)⋊6C8, C22⋊1(C4×C8), C4.155(C4×D4), (C2×C42).32C4, C23.30(C2×C8), (C2×C4).43C42, (C23×C4).27C4, C2.3(C4×M4(2)), C24.109(C2×C4), (C2×C4).90M4(2), (C22×C42).5C2, C22.27(C2×C42), C22.19(C22×C8), (C22×C8).469C22, (C23×C4).628C22, (C2×C42).987C22, C23.250(C22×C4), C22.36(C2×M4(2)), C2.2(C42.12C4), C4○3(C22.7C42), (C22×C4).1603C23, C22.7C42⋊42C2, C22.48(C42⋊C2), C42○2(C22.7C42), (C2×C4×C8)⋊5C2, C2.6(C2×C4×C8), (C2×C8)⋊27(C2×C4), (C2×C4).59(C2×C8), C2.2(C4×C22⋊C4), C2.2(C2×C22⋊C8), C42○(C2×C22⋊C8), (C2×C4).1493(C2×D4), (C2×C22⋊C8).48C2, (C2×C4).913(C4○D4), (C22×C4).433(C2×C4), (C2×C4).593(C22×C4), (C2×C4).395(C22⋊C4), C22.111(C2×C22⋊C4), (C2×C4)○2(C22.7C42), (C2×C4)○(C2×C22⋊C8), (C2×C42)○(C2×C22⋊C8), SmallGroup(128,480)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2 — C4×C22⋊C8
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C2×C4×C8 — C4×C22⋊C8
Lower central
C1 — C2 — C4×C22⋊C8
Upper central
C1 — C2×C42 — C4×C22⋊C8
Jennings
C1 — C2 — C2 — C22×C4 — C4×C22⋊C8

Generators and relations for C4×C22⋊C8
 G = < a,b,c,d | a4=b2=c2=d8=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 316 in 218 conjugacy classes, 120 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C23, C23, C23, C42, C42, C2×C8, C2×C8, C22×C4, C22×C4, C22×C4, C24, C4×C8, C22⋊C8, C2×C42, C2×C42, C2×C42, C22×C8, C23×C4, C23×C4, C22.7C42, C2×C4×C8, C2×C22⋊C8, C22×C42, C4×C22⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C23, C42, C22⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C4○D4, C4×C8, C22⋊C8, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C22×C8, C2×M4(2), C4×C22⋊C4, C2×C4×C8, C4×M4(2), C2×C22⋊C8, C42.12C4, C4×C22⋊C8

Smallest permutation representation of C4×C22⋊C8
►On 64 points
Generators in S64
(1 29 61 45)(2 30 62 46)(3 31 63 47)(4 32 64 48)(5 25 57 41)(6 26 58 42)(7 27 59 43)(8 28 60 44)(9 18 38 49)(10 19 39 50)(11 20 40 51)(12 21 33 52)(13 22 34 53)(14 23 35 54)(15 24 36 55)(16 17 37 56)

(2 21)(4 23)(6 17)(8 19)(10 44)(12 46)(14 48)(16 42)(26 37)(28 39)(30 33)(32 35)(50 60)(52 62)(54 64)(56 58)

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

80 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4X4Y···4AJ8A···8AF
order12···222224···44···48···8
size11···122221···12···22···2

80 irreducible representations

dim111111111222
type++++++
imageC1C2C2C2C2C4C4C4C8D4M4(2)C4○D4
kernelC4×C22⋊C8C22.7C42C2×C4×C8C2×C22⋊C8C22×C42C22⋊C8C2×C42C23×C4C22×C4C42C2×C4C2×C4
# reps12221164432484

Matrix representation of C4×C22⋊C8 ►in GL4(𝔽17) generated by

1000
01300
00130
00013
,
16000
01600
0010
00016
,
1000
0100
00160
00016
,
9000
0100
00013
00130
G:=sub<GL(4,GF(17))| [1,0,0,0,0,13,0,0,0,0,13,0,0,0,0,13],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[9,0,0,0,0,1,0,0,0,0,0,13,0,0,13,0] >;
 

C4×C22⋊C8 in GAP, Magma, Sage, TeX 

C_4\times C_2^2\rtimes C_8
 
% in TeX
 
G:=Group("C4xC2^2:C8");
 
// GroupNames label
 
G:=SmallGroup(128,480);
 
// by ID
 
G=gap.SmallGroup(128,480);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^2=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
 
// generators/relations
 

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