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G = C42.462C23  order 128 = 27

323rd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.462C23, C4.422+ (1+4), (C4×D8)⋊12C2, (C8×D4)⋊17C2, D4.Q8⋊6C2, D4⋊D4⋊9C2, D4⋊5D4⋊8C2, C8⋊8D4⋊36C2, C8⋊7D4⋊12C2, C4⋊C8⋊71C22, C4⋊C4.261D4, (C4×C8)⋊13C22, C22⋊D8⋊10C2, (C2×D4).234D4, D4.2D4⋊7C2, C2.45(D4○D8), (C4×D4)⋊24C22, C4.Q8⋊38C22, C2.D8⋊12C22, D4.18(C4○D4), C4⋊D4⋊15C22, C4⋊C4.228C23, C22⋊C8⋊64C22, (C2×C8).182C23, (C2×C4).489C24, C22⋊C4.101D4, (C22×C8)⋊13C22, Q8⋊C4⋊9C22, C23.107(C2×D4), C42.C2⋊8C22, D4⋊C4⋊60C22, (C2×SD16)⋊51C22, (C2×D8).138C22, (C2×D4).221C23, C22.11(C4○D8), C23.19D4⋊5C2, C23.48D4⋊7C2, (C2×Q8).206C23, C2.125(D4⋊5D4), C42⋊C2⋊21C22, C22⋊Q8.69C22, C23.24D4⋊10C2, C4.4D4.59C22, C22.749(C22×D4), (C22×C4).1133C23, C22.47C24⋊1C2, C42.78C22⋊4C2, (C22×D4).408C22, C2.59(C2×C4○D8), C4.214(C2×C4○D4), (C2×C4).166(C2×D4), (C2×D4⋊C4)⋊41C2, (C2×C4⋊C4).659C22, (C2×C4○D4).197C22, SmallGroup(128,2029)

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

Derived series
C1 — C2×C4 — C42.462C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — D4⋊5D4 — C42.462C23
Lower central
C1 — C2 — C2×C4 — C42.462C23
Upper central
C1 — C22 — C4×D4 — C42.462C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.462C23

Subgroups: 472 in 210 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×19], C8 [×4], C2×C4 [×5], C2×C4 [×14], D4 [×2], D4 [×15], Q8 [×2], C23 [×2], C23 [×8], C42, C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×5], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×2], D8 [×3], SD16, C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8, C4○D4 [×3], C24, C4×C8, C22⋊C8 [×2], D4⋊C4 [×7], Q8⋊C4 [×3], C4⋊C8, C4.Q8, C2.D8 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×3], C4×D4, C22≀C2, C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2, C22×C8 [×2], C2×D8 [×2], C2×SD16, C22×D4, C2×C4○D4, C2×D4⋊C4, C23.24D4, C8×D4, C4×D8, C22⋊D8, D4⋊D4, D4.2D4, C8⋊8D4, C8⋊7D4, D4.Q8, C23.19D4, C23.48D4, C42.78C22, D4⋊5D4, C22.47C24, C42.462C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C4○D8 [×2], C22×D4, C2×C4○D4, 2+ (1+4), D4⋊5D4, C2×C4○D8, D4○D8, C42.462C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=b2, ab=ba, cac-1=eae-1=a-1b2, ad=da, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, de=ed >

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)

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

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

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

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

Matrix representation ►G ⊆ GL4(𝔽17) generated by

4000
0400
00115
00116
,
0100
16000
0010
0001
,
51200
121200
00130
00134
,
16000
0100
00160
00016
,
4000
0400
00115
00016
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,1,1,0,0,15,16],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[5,12,0,0,12,12,0,0,0,0,13,13,0,0,0,4],[16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,4,0,0,0,0,1,0,0,0,15,16] >;
 

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A···4F4G4H4I4J4K4L4M4N8A8B8C8D8E···8J
order122222222224···44444444488888···8
size111122444882···24444888822224···4

35 irreducible representations

dim11111111111111112222244
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D82+ (1+4)D4○D8
kernelC42.462C23C2×D4⋊C4C23.24D4C8×D4C4×D8C22⋊D8D4⋊D4D4.2D4C8⋊8D4C8⋊7D4D4.Q8C23.19D4C23.48D4C42.78C22D4⋊5D4C22.47C24C22⋊C4C4⋊C4C2×D4D4C22C4C2
# reps11111111111111112114812

In GAP, Magma, Sage, TeX 

C_4^2._{462}C_2^3
 
% in TeX
 
G:=Group("C4^2.462C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2029);
 
// by ID
 
G=gap.SmallGroup(128,2029);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=a^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,a*d=d*a,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
 
// generators/relations
 

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