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G = 2+ 1+44C4order 128 = 27

3rd semidirect product of 2+ 1+4 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 3), monomial

Aliases: 2+ 1+44C4, 2- 1+43C4, C4○D4.42D4, (C2×D4).69D4, (C2×Q8).67D4, (C22×C8)⋊4C22, C23.31(C2×D4), (C22×C4).59D4, C4.113C22≀C2, C42⋊C228C2, D4.12(C22⋊C4), C42⋊C21C22, C22.11C22≀C2, C23.24D41C2, C23.7(C22⋊C4), C2.C25.2C2, Q8.12(C22⋊C4), C2.19(C243C4), (C2×M4(2))⋊40C22, C23.C231C2, (C22×C4).659C23, C4○D4.3(C2×C4), C4.7(C2×C22⋊C4), (C2×D4).65(C2×C4), (C2×C4).977(C2×D4), (C2×C4).5(C22×C4), (C2×Q8).58(C2×C4), (C2×C4○D4).7C22, (C22×C8)⋊C221C2, (C2×C4).11(C22⋊C4), C22.13(C2×C22⋊C4), SmallGroup(128,526)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — 2+ 1+44C4
C1C2C4C2×C4C22×C4C2×C4○D4C2.C25 — 2+ 1+44C4
C1C2C2×C4 — 2+ 1+44C4
C1C4C22×C4 — 2+ 1+44C4
C1C2C2C22×C4 — 2+ 1+44C4

Generators and relations for 2+ 1+44C4
 G = < a,b,c,d,e | a4=b2=e4=1, c2=d2=a2, bab=a-1, ebe-1=ac=ca, ad=da, eae-1=a2d, bc=cb, bd=db, dcd-1=a2c, ece-1=a2bd, ede-1=a >

Subgroups: 556 in 274 conjugacy classes, 66 normal (18 characteristic)
C1, C2, C2 [×9], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×15], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×28], D4 [×4], D4 [×28], Q8 [×4], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×4], C2×D4 [×20], C2×Q8, C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×8], C4○D4 [×36], C22⋊C8 [×2], C23⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×4], C42⋊C2 [×2], C22×C8, C2×M4(2), C2×C4○D4, C2×C4○D4 [×2], C2×C4○D4 [×6], 2+ 1+4 [×2], 2+ 1+4 [×4], 2- 1+4 [×2], 2- 1+4 [×2], (C22×C8)⋊C2, C23.C23, C23.24D4 [×2], C42⋊C22 [×2], C2.C25, 2+ 1+44C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, C22⋊C4 [×12], C22×C4, C2×D4 [×6], C2×C22⋊C4 [×3], C22≀C2 [×4], C243C4, 2+ 1+44C4

Smallest permutation representation of 2+ 1+44C4
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 2)(3 4)(5 6)(7 8)(9 12)(10 11)(13 28)(14 27)(15 26)(16 25)(17 18)(19 20)(21 32)(22 31)(23 30)(24 29)
(1 7 3 5)(2 8 4 6)(9 20 11 18)(10 17 12 19)(13 29 15 31)(14 30 16 32)(21 27 23 25)(22 28 24 26)
(1 12 3 10)(2 9 4 11)(5 19 7 17)(6 20 8 18)(13 30 15 32)(14 31 16 29)(21 28 23 26)(22 25 24 27)
(1 29 10 30)(2 14 9 13)(3 31 12 32)(4 16 11 15)(5 27 19 26)(6 22 18 23)(7 25 17 28)(8 24 20 21)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E···2J4A4B4C4D4E4F···4K4L4M4N4O8A8B8C8D8E8F
order122222···2444444···44444888888
size112224···4112224···48888444488

32 irreducible representations

dim1111111122224
type++++++++++
imageC1C2C2C2C2C2C4C4D4D4D4D42+ 1+44C4
kernel2+ 1+44C4(C22×C8)⋊C2C23.C23C23.24D4C42⋊C22C2.C252+ 1+42- 1+4C22×C4C2×D4C2×Q8C4○D4C1
# reps1112214424244

Matrix representation of 2+ 1+44C4 in GL4(𝔽17) generated by

16008
40131
01304
4001
,
1009
001316
04013
00016
,
10150
130416
10160
0140
,
13080
00113
0040
01310
,
82152
014512
52150
514514
G:=sub<GL(4,GF(17))| [16,4,0,4,0,0,13,0,0,13,0,0,8,1,4,1],[1,0,0,0,0,0,4,0,0,13,0,0,9,16,13,16],[1,13,1,0,0,0,0,1,15,4,16,4,0,16,0,0],[13,0,0,0,0,0,0,13,8,1,4,1,0,13,0,0],[8,0,5,5,2,14,2,14,15,5,15,5,2,12,0,14] >;

2+ 1+44C4 in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes_4C_4
% in TeX

G:=Group("ES+(2,2):4C4");
// GroupNames label

G:=SmallGroup(128,526);
// by ID

G=gap.SmallGroup(128,526);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,422,2019,1018,521,248,1411,124]);
// Polycyclic

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

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