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G = C4⋊2+ 1+4order 128 = 27

The semidirect product of C4 and 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

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

Aliases: C412+ 1+4, C22.85C25, C23.41C24, C42.573C23, C24.134C23, D4213C2, D415(C2×D4), (C2×D4)⋊39D4, C233(C2×D4), D42(C4⋊D4), C4⋊Q890C22, D46D421C2, D45D418C2, (C4×D4)⋊43C22, C233D47C2, (C2×C4).76C24, C2.31(D4×C23), C22≀C27C22, C4⋊D426C22, C4⋊C4.293C23, C41D450C22, (C2×C42)⋊58C22, (C23×C4)⋊42C22, C4.120(C22×D4), C22⋊Q831C22, (C2×D4).469C23, C4.4D481C22, (C22×D4)⋊36C22, C22⋊C4.20C23, (C2×Q8).445C23, C22.13(C22×D4), (C22×C4).358C23, (C2×2+ 1+4)⋊10C2, C22.D46C22, C2.31(C2×2+ 1+4), C2.22(C2.C25), C22.26C2434C2, C22.31C2415C2, (C2×C4×D4)⋊90C2, (C2×C4)⋊4(C2×D4), (C2×C4⋊D4)⋊67C2, (C2×C4⋊C4)⋊73C22, (C2×C4○D4)⋊29C22, (C2×C22⋊C4)⋊48C22, SmallGroup(128,2228)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C4⋊2+ 1+4
C1C2C22C23C24C23×C4C2×C4×D4 — C4⋊2+ 1+4
C1C22 — C4⋊2+ 1+4
C1C22 — C4⋊2+ 1+4
C1C22 — C4⋊2+ 1+4

Generators and relations for C4⋊2+ 1+4
 G = < a,b,c,d,e | a4=b4=c2=e2=1, d2=b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 1460 in 830 conjugacy classes, 430 normal (16 characteristic)
C1, C2 [×3], C2 [×18], C4 [×6], C4 [×15], C22, C22 [×10], C22 [×58], C2×C4 [×2], C2×C4 [×20], C2×C4 [×49], D4 [×16], D4 [×76], Q8 [×8], C23, C23 [×20], C23 [×36], C42 [×4], C22⋊C4 [×40], C4⋊C4 [×20], C22×C4 [×3], C22×C4 [×26], C22×C4 [×4], C2×D4 [×56], C2×D4 [×72], C2×Q8 [×4], C4○D4 [×48], C24 [×10], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×4], C4×D4 [×16], C22≀C2 [×16], C4⋊D4 [×40], C22⋊Q8 [×8], C22.D4 [×16], C4.4D4 [×4], C41D4 [×2], C4⋊Q8 [×2], C23×C4 [×2], C22×D4, C22×D4 [×20], C2×C4○D4 [×12], 2+ 1+4 [×16], C2×C4×D4, C2×C4⋊D4 [×4], C22.26C24 [×2], C233D4 [×4], C22.31C24 [×2], D42 [×4], D45D4 [×8], D46D4 [×4], C2×2+ 1+4 [×2], C4⋊2+ 1+4
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], 2+ 1+4 [×2], C25, D4×C23, C2×2+ 1+4, C2.C25, C4⋊2+ 1+4

Smallest permutation representation of C4⋊2+ 1+4
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 18 15 7)(2 17 16 6)(3 20 13 5)(4 19 14 8)(9 30 24 26)(10 29 21 25)(11 32 22 28)(12 31 23 27)
(1 8)(2 5)(3 6)(4 7)(9 31)(10 32)(11 29)(12 30)(13 17)(14 18)(15 19)(16 20)(21 28)(22 25)(23 26)(24 27)
(1 31 15 27)(2 32 16 28)(3 29 13 25)(4 30 14 26)(5 10 20 21)(6 11 17 22)(7 12 18 23)(8 9 19 24)
(1 27)(2 28)(3 25)(4 26)(5 21)(6 22)(7 23)(8 24)(9 19)(10 20)(11 17)(12 18)(13 29)(14 30)(15 31)(16 32)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D···2M2N···2U4A···4H4I···4V
order12222···22···24···44···4
size11112···24···42···24···4

44 irreducible representations

dim1111111111244
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D42+ 1+4C2.C25
kernelC4⋊2+ 1+4C2×C4×D4C2×C4⋊D4C22.26C24C233D4C22.31C24D42D45D4D46D4C2×2+ 1+4C2×D4C4C2
# reps1142424842822

Matrix representation of C4⋊2+ 1+4 in GL6(ℤ)

010000
-100000
001000
000100
000010
000001
,
010000
100000
00-1200
00-1100
000101
001-1-10
,
-100000
0-10000
00-1200
000100
000101
000-110
,
100000
010000
00-1020
000011
00-1010
001-1-10
,
-100000
0-10000
00-1020
000011
000010
0001-10

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

C4⋊2+ 1+4 in GAP, Magma, Sage, TeX

C_4\rtimes 2_+^{1+4}
% in TeX

G:=Group("C4:ES+(2,2)");
// GroupNames label

G:=SmallGroup(128,2228);
// by ID

G=gap.SmallGroup(128,2228);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,352,570,1684]);
// Polycyclic

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

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