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G = C4×2+ 1+4order 128 = 27

Direct product of C4 and 2+ 1+4

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

Aliases: C4×2+ 1+4, C22.15C25, C42.536C23, C24.474C23, C23.109C24, D4(C4×D4), Q8(C4×Q8), Q89(C22×C4), (C4×D4)⋊88C22, D410(C22×C4), C2.11(C24×C4), C4.41(C23×C4), C232(C22×C4), C4⋊C4.516C23, (C2×C4).161C24, (C23×C4)⋊16C22, (C2×C42)⋊41C22, (C4×Q8)⋊102C22, C22.5(C23×C4), (C2×D4).499C23, (C2×Q8).482C23, C42(C22.11C24), C42⋊C282C22, C22.11C2426C2, C2.3(C2×2+ 1+4), C22⋊C4.128C23, C2.2(C2.C25), (C22×C4).1295C23, (C22×D4).580C22, (C2×2+ 1+4).10C2, C42(C23.33C23), C23.33C2334C2, C4⋊C4(C4×D4), (C4×D4)(C4×D4), (C2×D4)(C4×D4), (C4×Q8)(C4×Q8), (C2×C4×D4)⋊68C2, C22⋊C4(C4×D4), (C4×C4○D4)⋊14C2, C4○D410(C2×C4), (C2×D4)⋊35(C2×C4), (C2×C4)⋊2(C22×C4), (C2×C4⋊C4)⋊121C22, (C2×C4)(C2×2+ 1+4), (C2×C22⋊C4)⋊78C22, (C2×C4)(C22.11C24), (C2×C4○D4).318C22, (C2×C4)(C23.33C23), SmallGroup(128,2161)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×2+ 1+4
C1C2C22C2×C4C22×C4C23×C4C2×C4×D4 — C4×2+ 1+4
C1C2 — C4×2+ 1+4
C1C2×C4 — 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, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d >

Subgroups: 1124 in 830 conjugacy classes, 686 normal (8 characteristic)
C1, C2 [×3], C2 [×18], C4 [×14], C4 [×15], C22, C22 [×18], C22 [×42], C2×C4, C2×C4 [×57], C2×C4 [×45], D4 [×72], Q8 [×8], C23 [×33], C23 [×12], C42 [×24], C22⋊C4 [×36], C4⋊C4 [×24], C22×C4 [×45], C22×C4 [×12], C2×D4 [×90], C2×Q8 [×2], C4○D4 [×48], C24 [×6], C2×C42 [×9], C2×C22⋊C4 [×18], C2×C4⋊C4 [×9], C42⋊C2 [×18], C4×D4 [×72], C4×Q8 [×8], C23×C4 [×6], C22×D4 [×9], C2×C4○D4 [×6], 2+ 1+4 [×16], C2×C4×D4 [×9], C4×C4○D4 [×6], C22.11C24 [×9], C23.33C23 [×6], C2×2+ 1+4, C4×2+ 1+4
Quotients: C1, C2 [×31], C4 [×16], C22 [×155], C2×C4 [×120], C23 [×155], C22×C4 [×140], C24 [×31], C23×C4 [×30], 2+ 1+4 [×2], C25, C24×C4, 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 17 11 16)(2 18 12 13)(3 19 9 14)(4 20 10 15)(5 25 30 24)(6 26 31 21)(7 27 32 22)(8 28 29 23)
(1 16)(2 13)(3 14)(4 15)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 17)(12 18)(21 31)(22 32)(23 29)(24 30)
(1 27 11 22)(2 28 12 23)(3 25 9 24)(4 26 10 21)(5 19 30 14)(6 20 31 15)(7 17 32 16)(8 18 29 13)
(1 24)(2 21)(3 22)(4 23)(5 17)(6 18)(7 19)(8 20)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D···2U4A4B4C4D4E···4AT
order12222···244444···4
size11112···211112···2

68 irreducible representations

dim111111144
type+++++++
imageC1C2C2C2C2C2C42+ 1+4C2.C25
kernelC4×2+ 1+4C2×C4×D4C4×C4○D4C22.11C24C23.33C23C2×2+ 1+42+ 1+4C4C2
# reps1969613222

Matrix representation of C4×2+ 1+4 in GL5(𝔽5)

20000
01000
00100
00010
00001
,
10000
02200
00300
00101
00240
,
40000
02200
01300
00101
04210
,
40000
04020
00031
04010
03420
,
40000
04020
00031
00010
00120

G:=sub<GL(5,GF(5))| [2,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,2,0,0,0,0,2,3,1,2,0,0,0,0,4,0,0,0,1,0],[4,0,0,0,0,0,2,1,0,4,0,2,3,1,2,0,0,0,0,1,0,0,0,1,0],[4,0,0,0,0,0,4,0,4,3,0,0,0,0,4,0,2,3,1,2,0,0,1,0,0],[4,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,2,3,1,2,0,0,1,0,0] >;

C4×2+ 1+4 in GAP, Magma, Sage, TeX

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

G:=Group("C4xES+(2,2)");
// GroupNames label

G:=SmallGroup(128,2161);
// by ID

G=gap.SmallGroup(128,2161);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,387,1123,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,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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