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G = C42.299C23  order 128 = 27

160th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.299C23, C4.1722+ (1+4), C8⋊6D4⋊42C2, C8⋊9D4⋊43C2, C4⋊C8⋊53C22, (C4×C8)⋊62C22, C22≀C2.7C4, C4⋊D4.26C4, C24.89(C2×C4), C8⋊C4⋊33C22, C22⋊Q8.26C4, C22⋊C8⋊47C22, (C2×C4).675C24, (C2×C8).436C23, (C22×C8)⋊57C22, (C4×D4).63C22, C24.4C4⋊38C2, C23.42(C22×C4), C2.30(Q8○M4(2)), (C2×M4(2))⋊49C22, C22.199(C23×C4), (C22×C4).942C23, (C23×C4).534C22, C22.D4.10C4, C22.19C24.14C2, C42⋊C2.87C22, C42.7C22⋊28C2, C2.49(C22.11C24), C4⋊C4.119(C2×C4), (C2×D4).143(C2×C4), C22⋊C4.20(C2×C4), (C2×C4).81(C22×C4), (C2×Q8).124(C2×C4), (C22×C8)⋊C2⋊34C2, (C22×C4).355(C2×C4), (C2×C4○D4).95C22, SmallGroup(128,1710)

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

Derived series
C1 — C22 — C42.299C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22.19C24 — C42.299C23
Lower central
C1 — C22 — C42.299C23
Upper central
C1 — C2×C4 — C42.299C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.299C23

Subgroups: 348 in 201 conjugacy classes, 124 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×19], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×7], Q8, C23, C23 [×4], C23 [×3], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×2], M4(2) [×6], C22×C4 [×2], C22×C4 [×6], C22×C4 [×2], C2×D4, C2×D4 [×6], C2×Q8, C4○D4 [×2], C24, C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×12], C4⋊C8 [×4], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C22×C8 [×2], C2×M4(2) [×6], C23×C4, C2×C4○D4, C24.4C4 [×2], (C22×C8)⋊C2 [×2], C42.7C22 [×2], C8⋊9D4 [×4], C8⋊6D4 [×4], C22.19C24, C42.299C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ (1+4) [×2], C22.11C24, Q8○M4(2) [×2], C42.299C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=dad=a-1, eae=ab2, bc=cb, bd=db, be=eb, dcd=ece=a2b2c, ede=b2d >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

115000000
016000000
016010000
016100000
00000100
000016000
000000016
00000010
,
40000000
04000000
00400000
00040000
000013000
000001300
000000130
000000013
,
101500000
001610000
701600000
741600000
00000010
00000001
000013000
000001300
,
115000000
016000000
1160160000
1161600000
00000100
00001000
00000001
00000010
,
160000000
161000000
160100000
000160000
000016000
00000100
000000160
00000001

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13],[1,0,7,7,0,0,0,0,0,0,0,4,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,1,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[16,16,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1] >;
 

38 conjugacy classes

class 1 2A2B2C2D···2H4A4B4C4D4E···4M8A···8P
order12222···244444···48···8
size11114···411114···44···4

38 irreducible representations

dim1111111111144
type++++++++
imageC1C2C2C2C2C2C2C4C4C4C42+ (1+4)Q8○M4(2)
kernelC42.299C23C24.4C4(C22×C8)⋊C2C42.7C22C8⋊9D4C8⋊6D4C22.19C24C22≀C2C4⋊D4C22⋊Q8C22.D4C4C2
# reps1222441444424

In GAP, Magma, Sage, TeX 

C_4^2._{299}C_2^3
 
% in TeX
 
G:=Group("C4^2.299C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1710);
 
// by ID
 
G=gap.SmallGroup(128,1710);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,891,675,1018,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b,a*b=b*a,c*a*c^-1=d*a*d=a^-1,e*a*e=a*b^2,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=e*c*e=a^2*b^2*c,e*d*e=b^2*d>;
 
// generators/relations
 

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