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G = M4(2).24C23  order 128 = 27

6th non-split extension by M4(2) of C23 acting via C23/C22=C2

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

Aliases: M4(2).24C23, C4○D4.48D4, D4○(C4.D4), (C2×C4).4C24, C24.14(C2×C4), Q8○(C4.10D4), Q8○M4(2)⋊11C2, (C22×D4).17C4, C4.134(C22×D4), D4.17(C22⋊C4), (C2×D4).355C23, C4.D4⋊19C22, Q8.17(C22⋊C4), C22.17(C23×C4), C23.61(C22×C4), (C2×Q8).328C23, C4.10D4⋊20C22, (C2×M4(2))⋊42C22, (C22×C4).273C23, (C2×2+ (1+4)).5C2, (C22×D4).319C22, M4(2).8C22⋊18C2, (C2×C4○D4).11C4, (C2×C4).443(C2×D4), C4.31(C2×C22⋊C4), (C2×D4).224(C2×C4), (C2×C4.D4)⋊28C2, (C22×C4).38(C2×C4), (C2×Q8).202(C2×C4), C22.4(C2×C22⋊C4), (C2×C4).244(C22×C4), (C2×C4○D4).82C22, C2.31(C22×C22⋊C4), SmallGroup(128,1620)

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

Derived series
C1 — C22 — M4(2).24C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2+ (1+4) — M4(2).24C23
Lower central
C1 — C2 — C22 — M4(2).24C23
Upper central
C1 — C2 — C2×C4○D4 — M4(2).24C23
Jennings
C1 — C2 — C2 — C2×C4 — M4(2).24C23

Subgroups: 724 in 378 conjugacy classes, 170 normal (9 characteristic)
C1, C2, C2 [×13], C4 [×8], C4 [×2], C22, C22 [×6], C22 [×27], C8 [×8], C2×C4, C2×C4 [×17], C2×C4 [×12], D4 [×12], D4 [×30], Q8 [×4], Q8 [×2], C23 [×9], C23 [×18], C2×C8 [×12], M4(2) [×8], M4(2) [×12], C22×C4 [×9], C2×D4 [×18], C2×D4 [×36], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×20], C24 [×6], C4.D4 [×12], C4.10D4 [×4], C2×M4(2) [×12], C8○D4 [×8], C22×D4 [×9], C2×C4○D4, C2×C4○D4 [×5], 2+ (1+4) [×8], C2×C4.D4 [×6], M4(2).8C22 [×6], Q8○M4(2) [×2], C2×2+ (1+4), M4(2).24C23

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C23×C4, C22×D4 [×2], C22×C22⋊C4, M4(2).24C23

Generators and relations
 G = < a,b,c,d,e | a8=b2=d2=e2=1, c2=a2b, bab=a5, cac-1=a5b, ad=da, ae=ea, cbc-1=a4b, bd=db, be=eb, cd=dc, ce=ec, ede=a4d >

Permutation representations
►On 16 points - transitive group 16T200
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 5)(3 7)(9 13)(11 15)

(1 10 7 12 5 14 3 16)(2 15 4 13 6 11 8 9)

(1 13)(2 14)(3 15)(4 16)(5 9)(6 10)(7 11)(8 12)

(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 9)(8 10)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(3,7)(9,13)(11,15), (1,10,7,12,5,14,3,16)(2,15,4,13,6,11,8,9), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,5)(3,7)(9,13)(11,15), (1,10,7,12,5,14,3,16)(2,15,4,13,6,11,8,9), (1,13)(2,14)(3,15)(4,16)(5,9)(6,10)(7,11)(8,12), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,9)(8,10) );
 
G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,5),(3,7),(9,13),(11,15)], [(1,10,7,12,5,14,3,16),(2,15,4,13,6,11,8,9)], [(1,13),(2,14),(3,15),(4,16),(5,9),(6,10),(7,11),(8,12)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,9),(8,10)])
 
G:=TransitiveGroup(16,200);
 

Matrix representation ►G ⊆ GL8(ℤ)

000000-10
00000001
0000-1000
00000100
00010000
00100000
01000000
10000000
,
-10000000
0-1000000
00-100000
000-10000
00001000
00000100
00000010
00000001
,
00001000
00000100
00000010
00000001
01000000
-10000000
00010000
00-100000
,
00010000
00-100000
0-1000000
10000000
00000001
000000-10
00000-100
00001000
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100

G:=sub<GL(8,Integers())| [0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
 

41 conjugacy classes

class 1 2A2B···2H2I···2N4A···4H4I4J8A···8P
order122···22···24···4448···8
size112···24···42···2444···4

41 irreducible representations

dim111111128
type+++++++
imageC1C2C2C2C2C4C4D4M4(2).24C23
kernelM4(2).24C23C2×C4.D4M4(2).8C22Q8○M4(2)C2×2+ (1+4)C22×D4C2×C4○D4C4○D4C1
# reps1662112481

In GAP, Magma, Sage, TeX 

M_{4(2)}._{24}C_2^3
 
% in TeX
 
G:=Group("M4(2).24C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1620);
 
// by ID
 
G=gap.SmallGroup(128,1620);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,521,2804,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^8=b^2=d^2=e^2=1,c^2=a^2*b,b*a*b=a^5,c*a*c^-1=a^5*b,a*d=d*a,a*e=e*a,c*b*c^-1=a^4*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=a^4*d>;
 
// generators/relations
 

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