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G = D4×C22⋊C4  order 128 = 27

Direct product of D4 and C22⋊C4

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

Aliases: D4×C22⋊C4, C25.22C22, C24.206C23, C23.220C24, C22.582+ (1+4), C2.1(D42), C22⋊5(C4×D4), C24⋊11(C2×C4), C24⋊3C4⋊7C2, (C2×D4).339D4, (C22×D4)⋊23C4, (C23×C4)⋊6C22, (D4×C23).9C2, C2.1(D4⋊5D4), (C2×C42)⋊15C22, C23.411(C2×D4), C23.7Q8⋊19C2, C22.99(C22×D4), C23.286(C4○D4), C23.23D4⋊10C2, C22.111(C23×C4), C23.213(C22×C4), C24.3C22⋊17C2, C2.C42⋊11C22, (C22×C4).1241C23, (C22×D4).607C22, C2.23(C22.11C24), (C2×C4×D4)⋊9C2, C2.28(C2×C4×D4), (C2×C4)⋊17(C2×D4), C4⋊1(C2×C22⋊C4), (C2×D4)⋊41(C2×C4), (C4×C22⋊C4)⋊36C2, (C22×C4)⋊28(C2×C4), C22⋊1(C2×C22⋊C4), (C2×C4⋊C4)⋊103C22, (C22×C22⋊C4)⋊8C2, (C2×C22⋊C4)⋊73C22, (C2×C4).452(C22×C4), C22.105(C2×C4○D4), C2.16(C22×C22⋊C4), (C2×D4)○(C2×C22⋊C4), SmallGroup(128,1070)

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

Derived series
C1 — C22 — D4×C22⋊C4
Chief series
C1 — C2 — C22 — C23 — C24 — C25 — D4×C23 — D4×C22⋊C4
Lower central
C1 — C22 — D4×C22⋊C4
Upper central
C1 — C23 — D4×C22⋊C4
Jennings
C1 — C23 — D4×C22⋊C4

Subgroups: 1292 in 626 conjugacy classes, 196 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×16], C4 [×4], C4 [×12], C22 [×3], C22 [×16], C22 [×88], C2×C4 [×12], C2×C4 [×48], D4 [×16], D4 [×24], C23, C23 [×22], C23 [×96], C42 [×4], C22⋊C4 [×4], C22⋊C4 [×26], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×12], C22×C4 [×24], C2×D4 [×24], C2×D4 [×44], C24, C24 [×12], C24 [×20], C2.C42 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×14], C2×C22⋊C4 [×8], C2×C4⋊C4 [×2], C4×D4 [×8], C23×C4, C23×C4 [×4], C22×D4 [×2], C22×D4 [×10], C22×D4 [×8], C25 [×2], C4×C22⋊C4, C24⋊3C4 [×2], C23.7Q8, C23.23D4 [×4], C24.3C22 [×2], C22×C22⋊C4 [×2], C2×C4×D4 [×2], D4×C23, D4×C22⋊C4

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×12], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×18], C4○D4 [×2], C24, C2×C22⋊C4 [×12], C4×D4 [×4], C23×C4, C22×D4 [×3], C2×C4○D4, 2+ (1+4) [×2], C22×C22⋊C4, C2×C4×D4, C22.11C24, D42 [×2], D4⋊5D4 [×2], D4×C22⋊C4

Generators and relations
 G = < a,b,c,d,e | a4=b2=c2=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Smallest permutation representation
►On 32 points
Generators in S32
(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)

(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)(13 16)(14 15)(17 18)(19 20)(21 24)(22 23)(25 28)(26 27)(29 30)(31 32)

(5 14)(6 15)(7 16)(8 13)(9 31)(10 32)(11 29)(12 30)

(1 23)(2 24)(3 21)(4 22)(5 14)(6 15)(7 16)(8 13)(9 31)(10 32)(11 29)(12 30)(17 28)(18 25)(19 26)(20 27)

(1 8 27 10)(2 5 28 11)(3 6 25 12)(4 7 26 9)(13 20 32 23)(14 17 29 24)(15 18 30 21)(16 19 31 22)
 
G:=sub<Sym(32)| (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), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32), (5,14)(6,15)(7,16)(8,13)(9,31)(10,32)(11,29)(12,30), (1,23)(2,24)(3,21)(4,22)(5,14)(6,15)(7,16)(8,13)(9,31)(10,32)(11,29)(12,30)(17,28)(18,25)(19,26)(20,27), (1,8,27,10)(2,5,28,11)(3,6,25,12)(4,7,26,9)(13,20,32,23)(14,17,29,24)(15,18,30,21)(16,19,31,22)>;
 
G:=Group( (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), (1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,30)(31,32), (5,14)(6,15)(7,16)(8,13)(9,31)(10,32)(11,29)(12,30), (1,23)(2,24)(3,21)(4,22)(5,14)(6,15)(7,16)(8,13)(9,31)(10,32)(11,29)(12,30)(17,28)(18,25)(19,26)(20,27), (1,8,27,10)(2,5,28,11)(3,6,25,12)(4,7,26,9)(13,20,32,23)(14,17,29,24)(15,18,30,21)(16,19,31,22) );
 
G=PermutationGroup([(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)], [(1,4),(2,3),(5,6),(7,8),(9,10),(11,12),(13,16),(14,15),(17,18),(19,20),(21,24),(22,23),(25,28),(26,27),(29,30),(31,32)], [(5,14),(6,15),(7,16),(8,13),(9,31),(10,32),(11,29),(12,30)], [(1,23),(2,24),(3,21),(4,22),(5,14),(6,15),(7,16),(8,13),(9,31),(10,32),(11,29),(12,30),(17,28),(18,25),(19,26),(20,27)], [(1,8,27,10),(2,5,28,11),(3,6,25,12),(4,7,26,9),(13,20,32,23),(14,17,29,24),(15,18,30,21),(16,19,31,22)])
 

Matrix representation ►G ⊆ GL6(ℤ)

-100000
0-10000
001000
000100
00001-2
00001-1
,
-100000
0-10000
001000
000100
00001-2
00000-1
,
100000
0-10000
001000
00-1-100
000010
000001
,
-100000
0-10000
00-1000
000-100
000010
000001
,
0-10000
-100000
00-1-200
001100
000010
000001

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,-2,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,-2,-1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-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,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

50 conjugacy classes

class 1 2A···2G2H···2S2T2U2V2W4A···4L4M···4Z
order12···22···222224···44···4
size11···12···244442···24···4

50 irreducible representations

dim11111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C4D4D4C4○D42+ (1+4)
kernelD4×C22⋊C4C4×C22⋊C4C24⋊3C4C23.7Q8C23.23D4C24.3C22C22×C22⋊C4C2×C4×D4D4×C23C22×D4C22⋊C4C2×D4C23C22
# reps112142221164842

In GAP, Magma, Sage, TeX 

D_4\times C_2^2\rtimes C_4
 
% in TeX
 
G:=Group("D4xC2^2:C4");
 
// GroupNames label
 
G:=SmallGroup(128,1070);
 
// by ID
 
G=gap.SmallGroup(128,1070);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,346]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^2=c^2=d^2=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,d*e=e*d>;
 
// generators/relations
 

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