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G = C4.152+ (1+4)  order 128 = 27

15th non-split extension by C4 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2

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

Aliases: C4.152+ (1+4), C8⋊D4⋊17C2, C8⋊8D4⋊27C2, Q8⋊D4⋊9C2, (C2×D4).155D4, (C2×Q8).131D4, C23.90(C2×D4), C2.D8⋊31C22, C4.Q8⋊35C22, C22⋊Q8⋊6C22, C22⋊SD16⋊10C2, C4⋊C4.139C23, C22⋊C8⋊18C22, (C2×C8).321C23, (C2×C4).398C24, (C22×C8)⋊37C22, C2.42(D4○SD16), Q8⋊C4⋊32C22, (C2×D4).149C23, C4⋊D4.39C22, (C2×Q8).136C23, (C22×Q8)⋊21C22, C42⋊C2⋊19C22, C23.20D4⋊25C2, C23.19D4⋊26C2, C2.79(C23⋊3D4), (C2×M4(2))⋊18C22, (C22×C4).301C23, (C2×SD16).81C22, C22.658(C22×D4), D4⋊C4.104C22, C22.29C24.14C2, (C22×D4).385C22, C23.38C23⋊14C2, (C2×C4).535(C2×D4), (C22×C8)⋊C2⋊14C2, (C2×C4○D4).166C22, SmallGroup(128,1932)

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

Derived series
C1 — C2×C4 — C4.152+ (1+4)
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C22.29C24 — C4.152+ (1+4)
Lower central
C1 — C2 — C2×C4 — C4.152+ (1+4)
Upper central
C1 — C22 — C2×C4○D4 — C4.152+ (1+4)
Jennings
C1 — C2 — C2 — C2×C4 — C4.152+ (1+4)

Subgroups: 476 in 205 conjugacy classes, 84 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×19], C8 [×4], C2×C4 [×4], C2×C4 [×14], D4 [×13], Q8 [×7], C23 [×3], C23 [×6], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×4], C2×C8, M4(2), SD16 [×4], C22×C4 [×3], C22×C4, C2×D4 [×3], C2×D4 [×2], C2×D4 [×7], C2×Q8, C2×Q8 [×2], C2×Q8 [×3], C4○D4 [×2], C24, C22⋊C8 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22≀C2 [×2], C4⋊D4 [×4], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4 [×2], C4⋊1D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×4], C22×D4, C22×Q8, C2×C4○D4, (C22×C8)⋊C2, Q8⋊D4 [×2], C22⋊SD16 [×2], C8⋊8D4 [×2], C8⋊D4 [×2], C23.19D4 [×2], C23.20D4 [×2], C22.29C24, C23.38C23, C4.152+ (1+4)

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ (1+4) [×2], C23⋊3D4, D4○SD16 [×2], C4.152+ (1+4)

Generators and relations
 G = < a,b,c,d,e | a4=c2=1, b4=e2=a2, d2=a-1b2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a-1b3, be=eb, dcd-1=ece-1=a2c, ede-1=ab2d >

Smallest permutation representation
►On 32 points
Generators in S32
(1 26 5 30)(2 27 6 31)(3 28 7 32)(4 29 8 25)(9 19 13 23)(10 20 14 24)(11 21 15 17)(12 22 16 18)

(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)

(2 25)(4 27)(6 29)(8 31)(9 13)(10 22)(11 15)(12 24)(14 18)(16 20)(17 21)(19 23)

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

115000000
116000000
016010000
1161600000
00001000
00000100
00000010
00000001
,
001070000
50070000
0125120000
1255120000
00007022
00003101315
00008870
000019310
,
10000000
01000000
101600000
100160000
00001000
00000100
00001010160
000007016
,
100150000
001160000
1160160000
100160000
0000161600
00002100
00001001616
000014721
,
101500000
001610000
101600000
1161600000
00001100
000001600
00001001616
000001001

G:=sub<GL(8,GF(17))| [1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,5,0,12,0,0,0,0,0,0,12,5,0,0,0,0,10,0,5,5,0,0,0,0,7,7,12,12,0,0,0,0,0,0,0,0,7,3,8,1,0,0,0,0,0,10,8,9,0,0,0,0,2,13,7,3,0,0,0,0,2,15,0,10],[1,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,10,0,0,0,0,0,0,1,10,7,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,1,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,0,16,2,10,14,0,0,0,0,16,1,0,7,0,0,0,0,0,0,16,2,0,0,0,0,0,0,16,1],[1,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,10,0,0,0,0,0,1,16,0,10,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1] >;
 

Character table of C4.152+ (1+4)

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D8E8F
 size 11114448822444888888444488
ρ111111111111111111111111111    trivial
ρ211111-1-11-1111-1-1-1-111-11-1-1-1-111    linear of order 2
ρ31111-1-111111-11-1-1-1-111-111-1-1-11    linear of order 2
ρ41111-11-11-111-1-1111-11-1-1-1-111-11    linear of order 2
ρ511111-1-11-1111-1-11-11-11-11111-1-1    linear of order 2
ρ611111111111111-111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ71111-11-11-111-1-11-11-1-11111-1-11-1    linear of order 2
ρ81111-1-111111-11-11-1-1-1-11-1-1111-1    linear of order 2
ρ91111-11-1-1111-1-111-111-1-111-1-11-1    linear of order 2
ρ101111-1-11-1-111-11-1-11111-1-1-1111-1    linear of order 2
ρ1111111-1-1-11111-1-1-11-11-111111-1-1    linear of order 2
ρ121111111-1-1111111-1-1111-1-1-1-1-1-1    linear of order 2
ρ131111-1-11-1-111-11-1111-1-1111-1-1-11    linear of order 2
ρ141111-11-1-1111-1-11-1-11-111-1-111-11    linear of order 2
ρ151111111-1-111111-1-1-1-1-1-1111111    linear of order 2
ρ1611111-1-1-11111-1-111-1-11-1-1-1-1-111    linear of order 2
ρ17222222-200-2-2-22-2000000000000    orthogonal lifted from D4
ρ182222-2-2-200-2-2222000000000000    orthogonal lifted from D4
ρ192222-22200-2-22-2-2000000000000    orthogonal lifted from D4
ρ2022222-2200-2-2-2-22000000000000    orthogonal lifted from D4
ρ214-44-4000004-4000000000000000    orthogonal lifted from 2+ (1+4)
ρ224-44-400000-44000000000000000    orthogonal lifted from 2+ (1+4)
ρ2344-4-40000000000000000002√-22√-200    complex lifted from D4○SD16
ρ2444-4-40000000000000000002√-22√-200    complex lifted from D4○SD16
ρ254-4-4400000000000000002√-22√-20000    complex lifted from D4○SD16
ρ264-4-4400000000000000002√-22√-20000    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4._{15}2_+^{(1+4)}
 
% in TeX
 
G:=Group("C4.15ES+(2,2)");
 
// GroupNames label
 
G:=SmallGroup(128,1932);
 
// by ID
 
G=gap.SmallGroup(128,1932);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,675,1018,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=c^2=1,b^4=e^2=a^2,d^2=a^-1*b^2,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^-1*b^3,b*e=e*b,d*c*d^-1=e*c*e^-1=a^2*c,e*d*e^-1=a*b^2*d>;
 
// generators/relations
 

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