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G = M4(2).25C23  order 128 = 27

7th non-split extension by M4(2) of C23 acting via C23/C22=C2

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

Aliases: M4(2).25C23, C4○D4.49D4, (C2×C4).5C24, Q8○(C4.D4), D4○(C4.10D4), Q8○M4(2)⋊12C2, C4.135(C22×D4), (C22×Q8).15C4, D4.18(C22⋊C4), (C2×D4).356C23, Q8.18(C22⋊C4), C23.98(C22×C4), C22.18(C23×C4), (C2×Q8).329C23, (C22×C4).274C23, C4.10D4.9C22, C4.D4.12C22, (C2×2- (1+4)).4C2, (C22×Q8).256C22, (C2×M4(2)).242C22, M4(2).8C22⋊19C2, (C2×C4○D4).12C4, (C2×C4).444(C2×D4), C4.32(C2×C22⋊C4), (C2×D4).225(C2×C4), (C22×C4).39(C2×C4), (C2×Q8).203(C2×C4), C22.5(C2×C22⋊C4), (C2×C4.10D4)⋊28C2, (C2×C4).109(C22×C4), (C2×C4○D4).83C22, C2.32(C22×C22⋊C4), SmallGroup(128,1621)

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

Derived series
C1 — C22 — M4(2).25C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2- (1+4) — M4(2).25C23
Lower central
C1 — C2 — C22 — M4(2).25C23
Upper central
C1 — C2 — C2×C4○D4 — M4(2).25C23
Jennings
C1 — C2 — C2 — C2×C4 — M4(2).25C23

Subgroups: 548 in 352 conjugacy classes, 170 normal (9 characteristic)
C1, C2, C2 [×9], C4 [×8], C4 [×6], C22, C22 [×6], C22 [×7], C8 [×8], C2×C4, C2×C4 [×21], C2×C4 [×24], D4 [×12], D4 [×14], Q8 [×4], Q8 [×18], C23 [×5], C2×C8 [×12], M4(2) [×8], M4(2) [×12], C22×C4 [×15], C2×D4, C2×D4 [×9], C2×Q8, C2×Q8 [×9], C2×Q8 [×20], C4○D4 [×8], C4○D4 [×36], C4.D4 [×4], C4.10D4 [×12], C2×M4(2) [×12], C8○D4 [×8], C22×Q8 [×5], C2×C4○D4, C2×C4○D4 [×9], 2- (1+4) [×8], C2×C4.10D4 [×6], M4(2).8C22 [×6], Q8○M4(2) [×2], C2×2- (1+4), M4(2).25C23

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).25C23

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

Smallest permutation representation
►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)

(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(26 30)(28 32)

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

22890009
22980090
89220900
98229000
0000151598
0004151589
0000981515
0400891515
,
10000000
01000000
00100000
00010000
1529916000
2159901600
9915200160
9921500016
,
1529915000
2159901500
9915200150
9921500015
000021588
100015288
000088215
001088152
,
04000000
130000000
000130000
00400000
900150400
082013000
028000013
150090040
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100

G:=sub<GL(8,GF(17))| [2,2,8,9,0,0,0,0,2,2,9,8,0,0,0,4,8,9,2,2,0,0,0,0,9,8,2,2,0,4,0,0,0,0,0,9,15,15,9,8,0,0,9,0,15,15,8,9,0,9,0,0,9,8,15,15,9,0,0,0,8,9,15,15],[1,0,0,0,15,2,9,9,0,1,0,0,2,15,9,9,0,0,1,0,9,9,15,2,0,0,0,1,9,9,2,15,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[15,2,9,9,0,1,0,0,2,15,9,9,0,0,0,0,9,9,15,2,0,0,0,1,9,9,2,15,0,0,0,0,15,0,0,0,2,15,8,8,0,15,0,0,15,2,8,8,0,0,15,0,8,8,2,15,0,0,0,15,8,8,15,2],[0,13,0,0,9,0,0,15,4,0,0,0,0,8,2,0,0,0,0,4,0,2,8,0,0,0,13,0,15,0,0,9,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0],[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···2H2I2J4A···4H4I···4N8A···8P
order122···2224···44···48···8
size112···2442···24···44···4

41 irreducible representations

dim111111128
type++++++-
imageC1C2C2C2C2C4C4D4M4(2).25C23
kernelM4(2).25C23C2×C4.10D4M4(2).8C22Q8○M4(2)C2×2- (1+4)C22×Q8C2×C4○D4C4○D4C1
# reps1662141281

In GAP, Magma, Sage, TeX 

M_{4(2)}._{25}C_2^3
 
% in TeX
 
G:=Group("M4(2).25C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1621);
 
// by ID
 
G=gap.SmallGroup(128,1621);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,456,521,2804,2028,124]);
 
// 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*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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